LIFTING HARMONIC MORPHISMS I: METRIZED COMPLEXES … · LIFTING HARMONIC MORPHISMS I: METRIZED COMPLEXES AND BERKOVICH SKELETA OMID AMINI, MATTHEW BAKER, ERWAN BRUGALLÉ, AND JOSEPH

LIFTING HARMONIC MORPHISMS I: METRIZED COMPLEXES AND BERKOVICH SKELETA

OMID AMINI, MATTHEW BAKER, ERWAN BRUGALLÉ, AND JOSEPH RABINOFF

ABSTRACT. Let K be an algebraically closed, complete non-Archimedean field. The purpose of this paperis to carefully study the extent to which finite morphisms of algebraic K-curves are controlled by certaincombinatorial objects, called skeleta. A skeleton is a metric graph embedded in the Berkovich analytifica-tion of X. A skeleton has the natural structure of a metrized complex of curves. We prove that a finitemorphism of K-curves gives rise to a finite harmonic morphism of a suitable choice of skeleta. We use thisto give analytic proofs of stronger “skeletonized” versions of some foundational results of Liu-Lorenzini,Coleman, and Liu on simultaneous semistable reduction of curves. We then consider the inverse problemof lifting finite harmonic morphisms of metrized complexes to morphisms of curves over K. We provethat every tamely ramified finite harmonic morphism of Λ-metrized complexes of k-curves lifts to a finitemorphism of K-curves. If in addition the ramification points are marked, we obtain a complete classifica-tion of all such lifts along with their automorphisms. This generalizes and provides new analytic proofsof earlier results of Saïdi and Wewers. As an application, we discuss the relationship between harmonicmorphisms of metric graphs and induced maps between component groups of Néron models, providing anegative answer to a question of Ribet motivated by number theory.

This article is the first in a series of two. The second article contains several applications of our liftingresults to questions about lifting morphisms of tropical curves.

Throughout this paper, unless explicitly stated otherwise, K denotes a complete algebraically closed non-Archimedean field with nontrivial valuation val : K → R ∪ ∞. Its valuation ring is denoted R, its maximalideal is mR, and the residue field is k = R/mR. We denote the value group of K by Λ = val(K×) ⊂ R.

1. INTRODUCTION

This article is the first in a series of two. The second, entitled Lifting harmonic morphisms II: tropicalcurves and metrized complexes, will be cited as [ABBR14]; references of the form “Theorem II.1.1” willrefer to Theorem 1.1 in [ABBR14].

1.1. The purpose of this paper is to carefully study the extent to which finite morphisms of algebraicK-curves are controlled by certain combinatorial objects, called skeleta. Let X be a smooth, proper,connected K-curve. Roughly speaking, a triangulation (X,V ∪ D) of X (with respect to a finiteset of punctures D ⊂ X(K)) is a finite set V of points in the Berkovich analytification Xan of Xwhose removal partitions Xan into open balls and finitely many open annuli (with the puncturesbelonging to distinct open balls). Triangulations of (X,D) are naturally in one-to-one correspondencewith semistable models X of (X,D): see Section 5. A triangulation (X,V ∪D) gives rise to a skeletonΣ(X,V ∪D) of X. The skeleton of a triangulated punctured curve is the dual graph of the special fiberXk of the corresponding semistable model, equipped with a canonical metric, along with completedinfinite rays in the directions of the punctures. There are many skeleta in Xan, although if X \ Dis hyperbolic (i.e., has negative Euler characteristic χ(X,D) = 2 − 2g(X) − #D), then there existsa unique minimal skeleton. A skeleton of (X,D) is by definition a subset Σ ⊂ Xan of the formΣ(X,V ∪D) for some triangulation of X.

1.2. Skeletal simultaneous semistable reduction. First we will prove that finite morphisms ofcurves induce morphisms of skeleta:

The authors thank Amaury Thuillier and Antoine Ducros for their help with some of the descent arguments in (5.2). Weare grateful to Andrew Obus for a number of useful comments based on a careful reading of the first arXiv version of thismanuscript. M.B. was partially supported by NSF grant DMS-1201473. E.B. was partially supported by the ANR-09-BLAN-0039-01.

1

arX

iv:1

303.

4812

v3 [

mat

h.A

G]

15

Apr

201

4

2 OMID AMINI, MATTHEW BAKER, ERWAN BRUGALLÉ, AND JOSEPH RABINOFF

Theorem A. Let ϕ : X ′ → X be a finite morphism of smooth, proper, connected K-curves and letD ⊂ X(K) be a finite set. There exists a skeleton Σ ⊂ X such that Σ′ := ϕ−1(Σ) is a skeleton of X ′. Forany such Σ the map ϕ : Σ′ → Σ is a finite harmonic morphism of metric graphs.

See Corollaries 4.18 and 4.28. Harmonic morphisms of graphs are defined in Section 2. Due tothe close relationship between semistable models and skeleta, Theorem A should be interpreted asa very strong simultaneous semistable reduction theorem. In fact, we will show how to formallyderive from Theorem A the simultaneous semistable reduction theorems of Liu–Lorenzini [LL99],Coleman [Col03], and Liu [Liu06]. Our version of these results hold over any non-Archimedean fieldK0, not assumed to be discretely valued. As an example, the following weak form of Liu’s theorem isa consequence of Theorem A:

Corollary. Let X,X ′ be smooth, proper, geometrically connected curves over a non-Archimedean field K0

and let ϕ : X ′ → X be a finite morphism. Then there exists a finite, separable extension K1 of K0 andsemistable models X,X′ of the curves XK1 , X

′K1

, respectively, such that ϕK1 extends to a finite morphismX′ → X.

We discuss simultaneous semistable reduction theorems in Section 5. We wish to emphasize thatTheorem A does not follow from any classical simultaneous semistable reduction theorem, in thatthere exist finite morphisms ϕ : X′ → X of semistable models such that the inverse image of thecorresponding skeleton of X is not a skeleton of X ′. (See, however, [CKK] where a skeletal versionof Liu’s theorem is derived from Liu’s method of proof in the discretely valued case.) Our proof ofTheorem A is entirely analytic, resting on an analysis of morphisms between open annuli and openballs; in particular it makes almost no reference to semistable reduction theory, and is therefore quitedifferent from Liu and Liu–Lorenzini’s approach.

A skeletal simultaneous semistable reduction theorem is important precisely when one wants toobtain a well-behaved morphism of graphs from a finite morphism of curves. This is crucial forobtaining the spectral lower bound on the gonality in [CKK], for instance.

1.3. Lifting harmonic morphisms. Our second goal is in a sense inverse to the first: we wish tolift finite harmonic morphisms of metric graphs to finite morphisms of curves. More precisely, let(X,D) be a punctured K-curve as above, let Σ be a skeleton, and let ϕ : Σ′ → Σ be a finite harmonicmorphism of metric graphs. It is natural to ask whether there exists a curve X ′ and a finite morphismϕ : X ′ → X such that ϕ−1(Σ) is a skeleton of X ′ and ϕ−1(Σ) ∼= Σ′ as metric graphs over Σ. In generalthe answer is “no”: there are subtle “Hurwitz obstructions” to such a lift, as described below. Onesolution is to enrich Σ,Σ′ with the extra structure of metrized complexes of curves. A metrized complexof curves is basically a metric graph Γ and for every (finite) vertex p ∈ Γ, the data of a smooth,proper k-curve Cp, along with an identification of the edges adjacent to p with distinct k-points of Cp.There is a notion of a finite harmonic morphism of metrized complexes of curves, which consists of afinite harmonic morphism ϕ : Γ′ → Γ of underlying metric graphs, and for every vertex p′ ∈ Γ′ withp = ϕ(p′), a finite morphism ϕp′ : Cp′ → Cp, satisfying various compatibility conditions.

Now let Σ = Σ(X,V ∪ D) for a triangulated punctured curve (X,V ∪ D). We will show that Σis naturally a metrized complex of curves, and that finite harmonic morphisms Σ′ → Σ of metrizedcomplexes do lift to finite morphisms of curves, under a mild tameness hypothesis. Moreover we havevery precise control over the set of such lifts entirely in terms of the morphism of metrized complexes.The main lifting theorem, stated somewhat imprecisely, is as follows:

Theorem B. (Lifting theorem) Let (X,V ∪D) be a triangulated puncturedK-curve, let Σ be its skeleton,and let Σ′ → Σ be a tame covering of metrized complexes of curves. Then there exists a curve X ′ anda finite morphism ϕ : X ′ → X, branched only over D, such that ϕ−1(Σ) is a skeleton of (X ′, ϕ−1(D))and ϕ−1(Σ) ∼= Σ′ as metrized complexes of curves over Σ. There are finitely many such lifts X ′ up toX-isomorphism. There are explicit descriptions of the set of lifts and the automorphism group of each lift(as a cover of X) in terms of the morphism Σ′ → Σ.

LIFTING HARMONIC MORPHISMS I 3

See Theorems 7.4, 7.7, and 7.10 for the precise statements. Tame coverings are defined in Defini-tion 2.21. The hardest part of the proof of Theorem B is a local lifting result, Theorem 6.18, in whichthe target curve X is replaced by a neighborhood of a vertex p ∈ Σ. Theorem 6.18 eventually reducesto classical results about the tamely ramified étale fundamental group.

It is worth mentioning that given a metrized complex of curves C with edge lengths contained inΛ, it is not hard to construct a triangulated punctured K-curve (X,V ∪D) such that Σ(X,V ∪D) ∼= C.See Theorem 3.24.

1.4. Our lifting results extend several theorems in the literature, as well as making them more pre-cise. Saïdi [Saï97, Théoréme 3.7] proves a version of the lifting theorem for semistable formal curves(without punctures) over a complete discrete valuation ring. His methods also make use of the tamelyramified étale fundamental group and analytic gluing arguments. Wewers [Wew99] works moregenerally with marked curves over a complete Noetherian local ring using deformation-theoretic ar-guments, proving that every tamely ramified admissible cover of a marked semistable curve over theresidue field of a complete Noetherian local ring lifts. Wewers classifies the possible lifts (fixing, as wedo, a lift of the target) in terms of certain non-canonical “deformation data” depending on compatiblechoices of formal coordinates at the nodes on both the source and target curves.

Our results have several advantages compared to these previous approaches. For one, we are ableto work over non-Noetherian rank-1 valuation rings, and descend the lifting theorems a postiori to anessentially arbitrary non-Archimedean field: see (7.11). The real novelty, however, is the systematicuse of metrized complexes of curves in the formulation of the lifting theorems, in particular theclassification of the set of lifts X ′ of Σ′ → Σ. For example, in our situation, Wewers’ non-canonicaldeformation data are replaced by certain canonical gluing data, which means that the automorphismgroup of the morphism of metrized complexes acts naturally on the set of gluing data (but not on theset of deformation data). This is what allows us to classify lifts up to isomorphism as covers of thetarget curve, as well as determine the automorphism group of such a lift. Saïdi’s lifting theorem ismore similar to ours, with generators of inertia groups over nodal points playing the role of gluingdata. It is not clear to the authors whether his methods can be extended to give a classification of lifts.

The question of lifting (and classification of all possible liftings) in the wildly ramified case is moresubtle and cannot be guaranteed in general. See Remark 7.6.

1.5. Homomorphisms of component groups. We will discuss in Section 8 the relationship betweenharmonic morphisms of metric graphs and component groups of Néron models, connecting our liftingresults to natural questions in arithmetic algebraic geometry. The idea is as follows. Let X be asmooth, proper, geometrically connected curve over a discretely valued field K0, and suppose that Xadmits a semistable model X over the valuation ring R0 of K0. Let J be the Jacobian of X and let ΦXbe the group of connected components of the special fiber of the Néron model of J .

Given a metric graph Γ with integer edge lengths, there is a naturally associated abelian groupJacreg(Γ), called its regularized Jacobian. The association Γ 7→ Jacreg(Γ) satisfies Picard and Albanesefuntoriality: that is, given a finite harmonic morphism Γ′ → Γ there are natural push-forward andpull-back homomorphisms Jacreg(Γ′)→ Jacreg(Γ) and Jacreg(Γ)→ Jacreg(Γ′).

A series of theorems of Raynaud, translated into our language, imply that ΦX ∼= Jacreg(ΣX) canon-ically, where ΣX is the skeleton induced by the semistable model X. We observe that this isomorphismrespects both Picard and Albanese functoriality.

For modular curves such as X0(N) with N prime, the minimal regular model has a special fiberconsisting of two projective lines intersecting transversely. Maps from such curves to elliptic curves,and their induced maps on component groups of Néron models, play an important role in arithmeticgeometry; for example, Ribet’s work establishing that the Shimura-Taniyama conjecture implies Fer-mat’s Last Theorem was motivated by the failure of such induced maps on component groups to besurjective in general. (The cokernel of such maps controls congruences between modular forms.) Ri-bet noticed that if one considers instead the case where the target curve has genus at least 2, then in


all examples he knew of, the induced map on component groups was surjective, and in personal com-munication with the second author, he asked whether this was a general property of curves whosereduction looks like that of X0(N). As a concrete application of Theorem B, the next propositionprovides a negative answer to this question.

Proposition C. There exists a finite morphism f : X ′ → X of semistable curves over a discretely valuedfield K0 with g(X) ≥ 2 such that:

• the special fiber of the minimal regular model of X ′ consists of two projective lines intersectingtransversely;

• the induced map f∗ : ΦX′ → ΦX on component groups of Néron models of Jacobians is notsurjective.

In order to prove Proposition C, we proceed as follows. As component groups can be calculated interms of metric graphs, one first finds a finite harmonic morphism of metric graphs ϕ : Γ′ → Γ suchthat Γ′ has two vertices and at least three edges, all of which have length one, and such that ϕ∗ :Jacreg(Γ′) → Jacreg(Γ) is not surjective. One then enriches ϕ to a morphism of metrized complexesof curves, with rational residue curves, and uses Theorem 3.24 and Theorem B (along with a descentargument) to lift this to a finite morphism of curves X ′ → X. Such a morphism satisfies the conditionsof Proposition C.

1.6. Applications to tropical lifting theorems. Tropical geometry has many applications to alge-braic geometry, in particular enumerative algebraic geometry. The basic strategy is to associate acombinatorial “tropical” object to an algebraic object, and prove that counting the former is somehowequivalent to counting the latter. An important ingredient in any such argument is a tropical liftingtheorem, which is a precise description of the algebraic lifts of a given tropical object.

Many of the intended applications of our lifting results are contained in the second paper [ABBR14],in which we prove a number of tropical lifting theorems. The skeleton of a curve, viewed as a metricgraph, is a tropical object associated to the curve; our lifting results can therefore be interpreted astropical lifting theorems for morphisms of metric graphs, in the following sense. Recall that Λ ⊂ R isthe value group of K. A Λ-metric graph is a metric graph Γ whose edge lengths are contained in Λ.We say that a finite harmonic morphism ϕ : Γ′ → Γ of Λ-metric graphs is liftable provided that thereexists a finite morphism of K-curves ϕ : X ′ → X, a set of punctures D ⊂ X(K), and a skeleton Σof (X,D) such that Σ ∼= Γ as metric graphs, ϕ−1(Σ) is a skeleton of (X ′, ϕ−1(D)), and Σ′ ∼= Γ′ asmetric graphs over Σ ∼= Γ. Theorem B (along with Theorem 3.24) shows that the obstruction to liftinga morphism of metric graphs ϕ : Γ′ → Γ to a morphism of curves is the same as the obstruction toenriching ϕ with the structure of a morphism of metrized complexes of curves. The following moreprecise statement is an immediate consequence of Proposition 7.15 and Theorem 3.24:

Corollary D. Assume char(k) = 0. Let ϕ : Γ′ → Γ be a finite harmonic morphism of Λ-metric graphs.Suppose that ϕ can be enriched to a morphism of metrized complexes of curves. Then ϕ is liftable.

We remark that we actually take advantage of the stronger form of Theorem B, in particular thecalculation of the automorphism group of a lift, when lifting group actions on metrized complexesin [ABBR14].

The problem of enriching a morphism of metric graphs to a morphism of metrized complexes isessentially equivalent to the problem of assigning k-curves Cp, Cp′ to vertices p ∈ Γ, p′ ∈ Γ′ and findingmorphisms of k-curves Cp′ → Cp with prescribed ramification profiles. The latter is a question aboutwhether certain Hurwitz numbers are nonzero. See Proposition II.3.3. Allowing arbitrary choices ofresidue curves Cp, Cp′ , this can always be done:

Theorem. Suppose that char(k) = 0. Any finite harmonic morphism ϕ : Γ′ → Γ of Λ-metric graphs isliftable to a morphism of curves.

See Theorem II.3.11. Imposing requirements on the lifted curves X,X ′ give rise to variants of thistropical lifting problem. The most basic such requirement is to specify the genera of the curves. An


augmented metric graph is a metric graph Γ along with an integer g(p) ∈ Z≥0 for every finite vertex pof Γ, called the genus. The genus of an augmented graph Γ is defined to be

g(Γ) = h1(Γ) +∑p

g(p),

where h1(Γ) is the first Betti number. Any metrized complex of curves has an underlying augmentedmetric graph, where g(p) := g(Cp), the genus of the residue curve. If Σ is a skeleton of a curve X,viewed as an augmented metric graph, then we have g(Σ) = g(X).

A finite harmonic morphism of augmented metric graphs Γ′ → Γ is liftable if the morphism ofunderlying metric graphs is liftable to a finite morphism of curves X ′ → X such that the isomorphismof Γ (resp. Γ′) with a skeleton of X (resp. X ′) respects the augmented metric graph structure. Inthis case, g(X) = g(Γ) and g(X ′) = g(Γ′). By our lifting theorem, Γ′ → Γ is liftable to a morphismof curves if and only if it can be enhanced to a morphism of metrized complexes such that g(Cp) =g(p) for every finite vertex. Now the obstruction to lifting is nontrivial: there exist finite harmonicmorphisms of metric graphs which cannot be enhanced to a morphism of metrized complexes. Thiscorresponds to the emptiness of certain Hurwitz spaces over k. See Section II.5 for the followingstriking example of this phenomenon:

Theorem. There exists an augmented metric graph Γ admitting a finite morphism of degree d > 0 to anaugmented metric graph T of genus zero, but which is not isomorphic to a skeleton of a d-gonal curve X.

In [ABBR14] we also give applications of the lifting theorem to the following kinds of tropicallifting problems, among others:

• When T is an augmented metric graph of genus zero, we study a variant of the lifting problemin which the genus of the source curve is prescribed, but the degree of the morphism is not.We use this to prove that linear equivalence of divisors on a tropical curve C coincides with theequivalence relation generated by declaring that the fibers of every finite harmonic morphismfrom C to the tropical projective line are equivalent.

• We study liftability of metrized complexes equipped with a finite group action, and as anapplication classify all (augmented) metric graphs arising as the skeleton of a hyperellipticcurve. As mentioned above, this study actually takes advantage of the more precise form ofTheorem B, not just the existence of a lift.

1.7. Organization of the paper. In Section 2 we recall some basic definitions of graphs with addi-tional structures, including metrized complexes of curves, and define harmonic morphisms betweenthem. We provide somewhat more detail than is strictly necessary for the exposition in this paper, butwill prove necessary in [ABBR14].

In Section 3 we review the structure theory of analytic curves, and define the skeleton of a curve.We show that the skeleton is naturally a metrized complex of curves, and we prove that any metrizedcomplex arises in this way. In Section 4 we develop “relative” versions of the results of Section 3.We prove that a finite morphism of curves induces a finite morphism between suitable choices ofskeleta, and that the map on skeleta is a finite harmonic morphism of metrized complexes of curves.In Section 5 we show how the previous two sections can be used to derive various simultaneoussemistable reduction theorems.

In Section 6 we prove a local lifting theorem, which essentially says that a tame covering ofmetrized complexes of curves can be lifted in a unique way to a tame covering of analytic K-curvesin a neighborhood of a vertex. This is the technical heart of the lifting theorem. In Section 7 weglobalize the considerations of the previous section, giving the classification of lifts of a tame coveringof metrized complexes of curves to a tame covering of triangulated punctured K-curves.

Section 8 contains an example application, in which we construct a morphism of curves over a dis-cretely valued field such that the covariantly associated morphism of component groups of Jacobianshas a prescribed behavior. We use this to answer a question of Ribet.


1.8. Related work. Another framework for the foundations of tropical geometry has been proposedby Kontsevich-Soibelman [KS01, KS06] and Mikhalkin [Mik06, Mik05], in which tropical objects areassociated to real one-parameter families of complex varieties. We refer to [KS01, KS06] for someconjectures relating this framework to Berkovich spaces. In this setting, the notion of metrized com-plex of curves is similar to the notion of phase-tropical curves, and Proposition 7.15 is a consequenceof Riemann’s Existence Theorem. We refer the interested reader to the forthcoming paper [Mik] formore details (see also [BBM11] where this statement is implicit, as well as Corollary II.3.8).

Harmonic morphisms of finite graphs were introduced by Urakawa in [Ura00] and further exploredin [BN09]. Harmonic morphisms of metric graphs have been introduced independently by severaldifferent people. Except for the integrality condition on the slopes, they appear already in Anand’spaper [Ana00]. The definition we use is the same as the one given in [Mik06, Cha12].

Finite harmonic morphisms of metrized complexes can be regarded as a metrized version of thenotion of admissible cover due to Harris and Mumford [HM82] (where, in addition, arbitrary ramifi-cation above smooth points is allowed). Recall that for two semistable curves Y ′ and Y over a fieldk, a finite surjective morphism ϕ : Y ′ → Y is an admissible cover if ϕ−1(Y sing) = Y ′sing and for eachsingular point y′ of Y ′, the ramification indices at y′ along the two branches intersecting at y′ coin-cide. (In addition, one usually imposes that all the other ramifications of ϕ are simple). An admissiblecover naturally gives rise to a finite harmonic morphism of metrized complexes: denoting by C theregularization of Y (the metrized complex associated to Y in which each edge has length one), defineC′ as the metrized complex obtained from Y ′ by letting the length of the edge associated to the doublepoint y′ be 1/ry′ (where ry′ is the ramification index of ϕ at y′ along either of the two branches). Themorphism ϕ of semistable curves naturally extends to a finite harmonic morphism ϕ : C′ → C: oneach edge e′ of C′ corresponding to a double point y′ of Y ′, the restriction of ϕ to e′ is linear withslope (or “expansion factor”) ry′ . Conversely, a finite harmonic morphism of metrized complexes ofcurves gives rise to an admissible cover of semistable curves (without the supplementary condition onsimple ramification) by forgetting the metrics on both sides, remembering only the expansion factoralong each edge.

A related but somewhat different Berkovich-theoretic point of view on simultaneous semistablereduction for curves can be found in Welliaveetil’s recent preprint [Wel13]; harmonic morphisms ofmetrized complexes of curves play an implicit role in his paper.

2. METRIC GRAPHS AND METRIZED COMPLEXES OF CURVES

In this section we recall several definitions of graphs with some additional structures and mor-phisms between them. A number of these definitions are now standard in tropical geometry; we referfor example to [BN07], [Mik06], [BBM11], and [AB12]. We also provide a number of examples.Some of the definitions in this section are not necessary in the generality or the form in which theyare presented for the purposes of this paper, but will be useful in [ABBR14].

Throughout this section, we fix Λ a non-trivial subgroup of (R,+).

2.1. Metric graphs. Given r ∈ Z≥1, we define Sr ⊂ C to be a “star with r branches”, i.e., a topologi-cal space homeomorphic to the convex hull in R2 of (0, 0) and any r points, no two of which lie on acommon line through the origin. We also define S0 = 0.

A finite topological graph Γ is the topological realization of a finite graph. That is to say, Γ is acompact 1-dimensional topological space such that for any point p ∈ Γ, there exists a neighborhoodUp of p in Γ homeomorphic to some Sr; moreover there are only finitely many points p with Uphomeomorphic to Sr with r 6= 2.

The unique integer r such that Up is homeomorphic to Sr is called the valence of p and denotedval(p). A point of valence different from 2 is called an essential vertex of Γ. The set of tangent directionsat p is Tp(Γ) = lim−→Up

π0(Up \ p), where the limit is taken over all neighborhoods of p isomorphic to

a star with r branches. The set Tp(Γ) has val(p) elements.


Definition 2.2. A metric graph is a finite connected topological graph Γ equipped with a completeinner metric on Γ\V∞(Γ), where V∞(Γ) ( Γ is some (finite) set of 1-valent vertices of Γ called infinitevertices of Γ. (An inner metric is a metric for which the distance between two points x and y is theminimum of the lengths of all paths between x and y.)

Let Γ be a metric graph containing a finite essential vertex. We say that Γ is a Λ-metric graph if thedistance between any two finite essential vertices of Γ lies in Λ. A Λ-point of Γ is a point of Γ whosedistance to any finite essential vertex of Γ lies in Λ.

Let Γ be a metric graph such that all essential vertices are infinite, so Γ is homeomorphic to acircle or a completed line. We say that Γ is a Λ-metric graph if it is a completed line or a circle whosecircumference is in Γ. Choose a point v ∈ Γ of valency equal to 2. A Λ-point of Γ is a point of Γ whosedistance to v lies in Λ.

One can equip Γ with a degenerate metric in which the infinite vertices are at infinite distance fromany other point of Γ. When no confusion is possible about the subgroup Λ, we will sometimes writesimply metric graph instead of Λ-metric graph.

Definition 2.3. Let Γ be a Λ-metric graph. A vertex set V (Γ) is a finite subset of the Λ-points of Γcontaining all essential vertices. An element of a fixed vertex set V (Γ) is called a vertex of Γ, and theclosure of a connected component of Γ \ V (Γ) is called an edge of Γ. An edge which is homeomorphicto a circle is called a loop edge. An edge adjacent to an infinite vertex is called an infinite edge. Wedenote by Vf (Γ) the set of finite vertices of Γ, and by Ef (Γ) the set of finite edges of Γ.

Fix a vertex set V (Γ). We denote by E(Γ) the set of edges of Γ. Since Γ is a metric graph, we canassociate to each edge e of Γ its length `(e) ∈ Λ∪+∞. Since the metric on Γ\V∞(Γ) is complete, anedge e is infinite if and only if `(e) = +∞. The notion of vertices and edges of Γ depends, of course,on the choice of a vertex set; we will fix such a choice without comment whenever there is no dangerof confusion.

Definition 2.4. Fix vertex sets V (Γ′) and V (Γ) for two Λ-metric graphs Γ′ and Γ, respectively, andlet ϕ : Γ′ → Γ be a continuous map.

• The map ϕ is called a (V (Γ′), V (Γ))-morphism of Λ-metric graphs if we have ϕ(V (Γ′)) ⊂ V (Γ),ϕ−1(E(Γ)) ⊂ E(Γ′), and the restriction of ϕ to any edge e′ of Γ′ is a dilation by some factorde′(ϕ) ∈ Z≥0.

• The map ϕ is called a morphism of Λ-metric graphs if there exists a vertex set V (Γ′) of Γ′ anda vertex set V (Γ) of Γ such that ϕ is a (V (Γ′), V (Γ))-morphism of Λ-metric graphs.

• The map ϕ is said to be finite if de′(ϕ) > 0 for any edge e′ ∈ E(Γ′).

An edge e′ of Γ′ is mapped to a vertex of Γ if and only if de′(ϕ) = 0. Such an edge is said to becontracted by ϕ. A morphism ϕ : Γ′ → Γ is finite if and only if there are no contracted edges, whichholds if and only if ϕ−1(p) is a finite set for any point p ∈ Γ. If ϕ is finite, then p′ ∈ Vf (Γ′) if and onlyif ϕ(p′) ∈ Vf (Γ).

The integer de′(ϕ) ∈ Z≥0 in Definition 2.4 is called the degree of ϕ along e′ (it is also sometimescalled the weight of e′ or expansion factor along e′ in the literature). Since `(ϕ(e′)) = de′(ϕ) · `(e′), itfollows in particular that if de′(ϕ) ≥ 1 then e′ is infinite if and only if ϕ(e′) is infinite. Let p′ ∈ V (Γ′), letv′ ∈ Tp′(Γ′), and let e′ ∈ E(Γ′) be the edge in the direction of v′. We define the directional derivativeof ϕ in the direction v′ to be dv′(ϕ) B de′(ϕ). If we set p = ϕ(p′), then ϕ induces a map

dϕ(p′) :v′ ∈ Tp′(Γ′) : dv′(ϕ) 6= 0

−→ Tp(Γ)

in the obvious way.

Example 2.5. Figure 1 depicts four examples of morphisms of metric graphs ϕ : Γ′ → Γ. We use thefollowing conventions in our pictures: black dots represent vertices of Γ′ and Γ; we label an edge withits degree if and only if the degree is different from 0 and 1; we do not specify the lengths of edges ofΓ′ and Γ.


The morphisms depicted in Figure 1(a), (b), and (c) are finite, while the one depicted in Figure1(d) is not.

d

ϕ

d

ϕ

2

2

ϕ

d d

p′ ϕ

a) b) c) d)

FIGURE 1

Definition 2.6. Let ϕ : Γ′ → Γ be a morphism of Λ-metric graphs, let p′ ∈ Γ′, and let p = ϕ(p′). Themorphism ϕ is harmonic at p′ provided that, for every tangent direction v ∈ Tp(Γ), the number

dp′(ϕ) :=∑

v′∈Tp′ (Γ′)

v′ 7→v

dv′(ϕ)

is independent of v. The number dp′(ϕ) is called the degree of ϕ at p′.We say that ϕ is harmonic if it is surjective and harmonic at all p′ ∈ Γ′; in this case the number

deg(ϕ) =∑p′ 7→p dp′(ϕ) is independent of p ∈ Γ, and is called the degree of ϕ.

Remark 2.7. In the above situation, if Γ consists of a single vertex v and ϕ : Γ′ → Γ is a morphism ofmetric graphs, then ϕ is by definition harmonic. In this case the quantity dp′(ϕ) is not defined, so weinclude the choice of a positive integer dp′(ϕ) for each p′ ∈ Vf (Γ′) as part of the data of the morphismϕ. Note that if ϕ is finite and Γ = p, then Γ′ = p′.

If both Γ′ and Γ have at least one edge, then a non-constant morphism which is harmonic at allp′ ∈ V (Γ′) is automatically surjective.

Example 2.8. The morphism depicted in Figure 1(a) is not harmonic, while the ones depicted inFigure 1(b), (c), and (d) are (for suitable choices of lengths).

2.9. Harmonic morphisms and harmonic functions. Given a metric graph Γ and a non-emptyopen set U in Γ, a function f : U → R is said to be harmonic on U if f is piecewise affine with integerslopes and for all x ∈ U , the sum of the slopes of f along all outgoing tangent directions at x is equalto 0.

Since we will not use it elsewhere in the paper, we omit the proof of the following result (which isvery similar to the proof of [BN09, Proposition 2.6]):

Proposition 2.10. A morphism ϕ : Γ′ → Γ of metric graphs is harmonic if and only if for every openset U ⊆ Γ and every harmonic function f : U → R, the pullback function ϕ∗f : ϕ−1(U) → R is alsoharmonic.

Equivalently, a morphism of metric graphs is harmonic if and only if germs of harmonic functionspull back to germs of harmonic functions.

2.11. For a metric graph Γ, we let Div(Γ) denote the free abelian group on Γ. Given a harmonicmorphism ϕ : Γ′ → Γ of metric graphs, there are natural pull-back and push-forward homomorphismsϕ∗ : Div(Γ)→ Div(Γ′) and ϕ∗ : Div(Γ′)→ Div(Γ) defined by

ϕ∗(p) =∑p′ 7→p

dp′(ϕ) (p′) and ϕ∗(p′) = (ϕ(p′))

and extending linearly. It is clear that for D ∈ Div(Γ) we have deg(ϕ∗(D)) = deg(ϕ) · deg(D) anddeg(ϕ∗(D)) = deg(D).


2.12. Augmented metric graphs. Here we consider Λ-metric graphs together with the data of anon-negative integer at each point. For a finite point p, this integer should be thought as the genus ofa “virtual algebraic curve” lying above p.

Definition 2.13. An augmented Λ-metric graph is a Λ-metric graph Γ along with a function g : Γ →Z≥0, called the genus function, such that g(p) = 0 for all points of Γ except possibly for finitely many(finite) Λ-points p ∈ Γ. The essential vertices of Γ are the points p for which val(p) 6= 2 or g(p) > 0. Avertex set of Γ is a vertex set V (Γ) of the underlying metric graph which contains all essential verticesof Γ as an augmented graph.

An augmented metric graph is said to be totally degenerate if the genus function is identically zero.The genus of Γ is defined to be

g(Γ) = h1(Γ) +∑p∈Γ

g(p),

where h1(Γ) = dimQH1(Γ,Q) is the first Betti number of Γ. The canonical divisor on Γ is

(2.13.1) KΓ =∑p∈Γ

(val(p) + 2g(p)− 2) (p).

A harmonic morphism of augmented Λ-metric graphs ϕ : Γ′ → Γ is a map which is a harmonicmorphism between the underlying metric graphs of Γ′ and Γ.

Note that both summations in the above definition are in fact over essential vertices of Γ. Thedegree of the canonical divisor of an augmented Λ-metric graph Γ is deg(KΓ) = 2g(Γ) − 2. Anaugmented metric graph of genus 0 will also be called a rational augmented metric graph.

2.14. Let ϕ : Γ′ → Γ be a harmonic morphism of augmented Λ-metric graphs. The ramification divisorof ϕ is the divisor R =

∑Rp′(p

′), where

(2.14.1) Rp′ = dp′(ϕ) ·(2− 2g(ϕ(p′))

)−(2− 2g(p′)

)−

∑v′∈Tp′ (Γ′)

(dv′(ϕ)− 1

).

Note that if p′ ∈ V∞(Γ′), then Rp′ ≥ 0 if dp′(ϕ) > 0, and Rp′ = −1 if dp′(ϕ) = 0. The definitionof Rv′ in (2.14.1) is rigged so that we have the following graph-theoretic analogue of the Riemann–Hurwitz formula:

(2.14.2) KΓ′ = ϕ∗(KΓ) +R.

In particular, the sum R =∑Rp′(p

′) is in fact finite.

Definition 2.15. Let ϕ : Γ′ → Γ be a finite harmonic morphism of augmented Λ-metric graphs. Wesay that ϕ is étale at a point p′ ∈ Γ′ provided that Rp′ = 0. We say that ϕ is generically étale if R issupported on the set of infinite vertices of Γ′ and that ϕ is étale if R = 0.

2.16. Metrized complexes of curves. Recall that k is an algebraically closed field. A metrizedcomplex of curves is, roughly speaking, an augmented metric graph Γ together with a chosen vertexset V (Γ) and a marked algebraic k-curve of genus g(p) for each finite vertex p ∈ V (Γ). More precisely:

Definition 2.17. A Λ-metrized complex of k-curves consists of the following data:

• An augmented Λ-metric graph Γ equipped with the choice of a distinguished vertex set V (Γ).• For every finite vertex p of Γ, a smooth, proper, connected k-curve Cp of genus g(p).• An injective function redp : Tp(Γ) → Cp(k), called the reduction map. We call the image of

redp the set of marked points on Cp.

Example 2.18. Figure 2 depicts a particular metrized complex of curves over C. At each finite vertexp there is an associated compact Riemann surface Cp, together with a finite set of marked points (inred).

Definition 2.19. A harmonic morphism of metrized complexes of curves consists of a harmonic(V (Γ′), V (Γ))-morphism ϕ : Γ′ → Γ of augmented metric graphs, and for every finite vertex p′ of Γ′


FIGURE 2

with dp′(ϕ) > 0 a finite morphism of curves ϕp′ : C ′p′ → Cϕ(p′), satisfying the following compatibilityconditions:

(1) For every finite vertex p′ ∈ V (Γ′) and every tangent direction v′ ∈ Tp′(Γ′) with dv′(ϕ) > 0,we have ϕp′(redp′(v

′)) = redϕ(p′)(dϕ(p′)(v′)), and the ramification degree of ϕp′ at redp′(v′)

is equal to dv′(ϕ).(2) For every finite vertex p′ ∈ V (Γ′) with dp′(ϕ) > 0, every tangent direction v ∈ Tϕ(p′)(Γ), and

every point x′ ∈ ϕ−1p′ (redϕ(p′)(v)) ⊂ C ′p′(k), there exists v′ ∈ Tp′(Γ′) such that redp′(v

′) = x′.(3) For every finite vertex p′ ∈ V (Γ′) with dp′(ϕ) > 0 we have dp′(ϕ) = deg(ϕp′).

A harmonic morphism of metrized complexes of curves is called finite if the underlying harmonicmorphism of (augmented) metric graphs is finite. Metrized complexes of curves with harmonic mor-phisms between them form a category, with finite harmonic morphisms giving rise to a subcategory.

If Γ has at least one edge, then (3) follows from (1) and (2), since the sum of the ramificationdegrees of a finite morphism of curves along any fiber is equal to the degree of the morphism.

Let ϕ : C ′ → C be a finite morphism of smooth, proper, connected k-curves. Recall that ϕ is tamelyramified if either char(k) = 0 or the ramification degree of ϕ at every closed point of C ′ is prime tothe characteristic of k.

Remark 2.20.

(1) Let ϕ : C ′ → C be a finite morphism. Let D be the (finite) set of branch points, let D′ =ϕ−1(D) ⊂ C ′(k), and let U = C \D and U ′ = C ′ \D′. Then ϕ is tamely ramified if and onlyif ϕ|U ′ : U ′ → U is a tamely ramified cover of U over C relative to D in the sense of [SGA1,XIII.2.1.3]: see the discussion in §2.0 of loc. cit.

(2) A tamely ramified morphism is separable (i.e. generically étale).(3) The Riemann–Hurwitz formula applies to a tamely ramified morphism.

Definition 2.21. Let ϕ : C′ → C be a finite harmonic morphism of metrized complexes of curves. Wesay that ϕ is a tame harmonic morphism if either char(k) = 0, or char(k) = p > 0 and ϕp′ is tamelyramified for all finite vertices p′ ∈ V (Γ′). We call ϕ a tame covering if in addition ϕ : Γ′ → Γ is agenerically étale finite morphism of augmented metric graphs.

Let ϕ : C′ → C be a tame harmonic morphism. Then each ϕp′ : Cp′ → Cϕ(p′) is a separablemorphism. Note that if char(k) = p then de′(ϕ) is prime to p for all edges e′ of Γ′. We have thefollowing kind of converse statement:

Proposition 2.22. Let ϕ : C′ → C be a finite harmonic morphism of metrized complexes of curves.Suppose that ϕ is generically étale and that ϕp′ : Cp′ → Cp is a separable morphism of curves for allfinite vertices p′ of Γ′. Let p′ be a finite vertex of Γ′, and suppose that ϕp′ is ramified at a point x′ ∈ C ′p′(k).Then there exists v′ ∈ Tp′(Γ′) such that redp′(v

′) = x′. In particular, if char(k) = p > 0 and de′(ϕ) isprime to p for all edges e′ of C′, then ϕp′ is tamely ramified and ϕ is a tame covering.

Proof. Let p = ϕ(p′). Since ϕ is generically étale we have

0 = Rp′ = dp′(ϕ)(2− 2g(p))− (2− 2g(p′))−∑

v′∈Tp′ (Γ′)

(dv′(ϕ)− 1).


Since g(p) = g(Cp), g(p′) = g(C ′p′), and dp′(ϕ) is the degree of ϕp′ : C ′p′ → Cp, this gives∑v′∈Tp′ (Γ′)

(dv′(ϕ)− 1) = deg(ϕp′)(2− 2g(Cp))− (2− 2g(C ′p′)).

The Riemann–Hurwitz formula as applied to ϕp′ says that the right-hand side of the above equationis equal to the degree of the ramification divisor of ϕp′ (this holds whenever ϕp′ is separable), so theproposition follows because for all v′ ∈ Tp′(Γ′), the ramification degree of ϕp′ at redp′(v

′) is dv′(ϕ). n

Remark 2.23. Loosely speaking, Proposition 2.22 says that all ramification of all morphisms ϕp′ is“visible” in the edge degrees of the underlying morphism of metric graphs. It follows from Proposi-tion 2.22 that if Γ′ = p′, then ϕp′ is étale.

3. ANALYTIC CURVES AND THEIR SKELETA

In this section we recall the definition and basic properties of the skeleton of the analytificationof an algebraic curve as outlined in [Tem] and elaborated in [BPR11, §5]. These foundational re-sults are well-known to experts, and appear in various guises in the literature. See for exampleBerkovich [Ber90, Ber99], Thuillier [Thu05], and Ducros [Duc08]. In this section we also show thatthe skeleton is naturally a metrized complex of curves, and prove that any metrized complex arises asthe skeleton of a curve. Finally, we introduce the important notion of a triangulated punctured curve,which is essentially a punctured curve along with the data of a skeleton with a distinguished set ofvertices.

Recall that K is an algebraically closed field which is complete with respect to a nontrivial non-Archimedean valuation val : K → R∪∞. Its valuation ring is R, its maximal ideal is mR, its residuefield is k = R/mR, and its value group is Λ = val(K×) ⊂ R.

By an analytic curve we mean a strictly K-analytic space X of pure dimension 1. For our purposesthe most important example of an analytic curve is the analytification of a smooth, connected, projec-tive algebraic K-curve. If X is an analytic curve, then we define H(X) = X \ X(K) as in [BPR11,3.10].

3.1. Analytic building blocks. We identify the set underlying the analytification of the affine linewith the set of multiplicative seminorms ‖ · ‖ : K[t] → R≥0 which extend the absolute value on K.We let val : A1,an → R ∪ ∞ denote the valuation map

val(x) = − log |x| = − log ‖t‖x.For a ∈ K× the standard closed ball of radius |a| is the affinoid domain

B(a) = val−1([val(a),∞]) ⊂ A1,an

and the standard open ball of radius |a| is the open analytic domain

B(a)+ = val−1((val(a),∞]) ⊂ A1,an.

Note that scaling by a gives isomorphisms B(1)∼−→ B(a) and B(1)+

∼−→ B(a)+. We call B(1) (resp.B(1)+) the standard closed (resp. open) unit ball. For a ∈ K× with val(a) ≥ 0 the standard closedannulus of modulus val(a) is the affinoid domain

S(a) = val−1([0, val(a)]) ⊂ A1,an

and if val(a) > 0 the standard open annulus of modulus val(a) is the open analytic domain

S(a)+ = val−1((0, val(a))) ⊂ A1,an.

The standard punctured open ball is the open analytic domain

S(0)+ = val−1((0,∞)) = B(1)+ \ 0 ⊂ A1,an.

By a closed unit ball (resp. open unit ball, resp. closed annulus, resp. open annulus, resp. puncturedopen ball) we will mean a K-analytic space which is isomorphic to the standard closed unit ball


(resp. the standard open unit ball, resp. the standard closed annulus of modulus val(a) for somea ∈ R \ 0, resp. the standard open annulus of modulus val(a) for some a ∈ mR \ 0, resp. thestandard punctured open ball). A generalized open annulus is a K-analytic space which is either anopen annulus or a punctured open ball. It is a standard fact that the modulus of a closed (resp. open)annulus is an isomorphism invariant; see for instance [BPR11, Corollary 5.6]. We define the modulusof a punctured open ball to be∞.

Let a ∈ mR (so a = 0 is allowed). There is a natural section σ : (0, val(a))→ S(a)+ of the valuationmap val : S(a)+ → (0, val(a)) sending r to the maximal point of the affinoid domain val−1(r) if r ∈ Λ,resp. the only point of val−1(r) if r /∈ Λ. The skeleton of S(a)+ is defined to be the image of σ and isdenoted Σ(S(a)+) = σ((0, val(a))). We will always identify the skeleton of the standard generalizedopen annulus S(a)+ with the interval or ray (0, val(a)). It follows from [BPR11, Proposition 5.5] thatthe skeleton of a standard open annulus or standard punctured open ball is an isomorphism invariant,so if A is a generalized open annulus, then we can define the skeleton Σ(A) of A to be the image of theskeleton of a standard open annulus or standard punctured open ball S(a)+ under any isomorphismS(a)+

∼−→ A.Let T be a metric space and let x, y ∈ T . A geodesic segment from x to y is the image of a locally

isometric embedding [a, b] → T with a 7→ x and b 7→ y. We often identify a geodesic segment with itsimage in T . Recall that an R-tree is a metric space T with the following properties:

(i) For all x, y ∈ T there is a unique geodesic segment [x, y] from x to y.(ii) For all x, y, z ∈ T , if [x, y] ∩ [y, z] = y, then [x, z] = [x, y] ∪ [y, z].

See [BR10, Appendix B]. It is proved in §1.4 of loc. cit. that H(B(1)) is naturally an R-tree. Sinceany path-connected subspace of an R-tree is an R-tree as well, if X is a standard open annulus orstandard (punctured) open ball then H(X) is an R-tree. It is proved in [BPR11, Corollary 5.61] thatthe metric structure on H(X) is an isomorphism invariant, so the same statement applies to open ballsand generalized open annuli. For a ∈ mR the section σ : (0, val(a))→ S(a)+ is an isometry.

It also follows from the results in [BR10, §1.4] that for any x ∈ H(B(1)) and any type-1 pointy ∈ B(1), there is a unique continuous injection α : [0,∞] → B(1) with α(0) = x and α(∞) = y,α([0,∞)) ⊂ H(B(1)), and such that α is an isometry when restricted to [0,∞). We let [x, y] denotethe image of α and we call [x, y] the geodesic segment from x to y. Similarly, if x and y are bothtype-1 points then there is a unique continuous injection α : [−∞,∞] → B(1) with α(−∞) = x andα(∞) = y, α(R) ⊂ H(B(1)), and such that α|R is an isometry; the image of α is called the geodesicsegment from x to y and is denoted [x, y]. Restricting to a suitable analytic subdomain of B(1) allowsus to define geodesic segments between any two points of an open ball or generalized open annulus.

3.2. Open balls. The closure of B(1)+ in B(1) consists of B(1)+ and a single type-2 point x, calledthe end of B(1)+. The end is the Shilov boundary point of B(1): see for instance the proof of [BPR11,Lemma 5.16]. The valuation map val : B(1)+ → (0,∞] extends to a continuous map val : B(1)+ ∪x → [0,∞]. The set B(1)+ ∪ x is path-connected and compact, being a closed subspace of thecompact space B(1). (In fact B(1)+ ∪ x is the one-point compactification of B(1)+.) For anyy ∈ B(1)+ the geodesic segment [x, y] ⊂ B(1) is contained in B(1)+ ∪ x.Lemma 3.3. Let X = Spec(A) be an irreducible affine K-curve and let ϕ : B(1)+ → Xan be amorphism with finite fibers. Then ϕ extends in a unique way to a continuous map B(1)+ ∪ x → Xan,and the image of x is a type-2 point of Xan.

Proof. Let f ∈ A be nonzero and define F : (0,∞) → R by F (r) = − log |f ϕ(σ(r))|, whereσ : (0,∞) → S(0)+ ⊂ B(1)+ is the canonical section of val. By [BPR11, Proposition 5.10], F is apiecewise affine function which is differentiable away from finitely many points r ∈ Λ ∩ (0,∞), andfor any point r ∈ (0,∞) at which F is differentiable, the derivative of F is equal to the number ofzeros y of f ϕ with val(y) > r. It follows that F (Λ∩ (0,∞)) ⊂ Λ. Since f ϕ has finitely many zeros,the limit limr→0 F (r) exists and is contained in Λ. We define ‖f‖0 = exp(− limr→0 F (r)). It is easy tosee that f 7→ ‖f‖0 is a multiplicative seminorm on A extending the absolute value on K, so ‖ · ‖0 is


a point in Xan. Define ϕ(x) = ‖ · ‖0. One shows as in the proof of [BPR11, Lemma 5.16] that ϕ iscontinuous on B(1)+ ∪ x.

It remains to show that ϕ(x) = ‖ · ‖0 is a type-2 point of Xan. Let f ∈ A be a non-constantfunction that has a zero on ϕ(B(1)+). Since − log ‖f‖0 ∈ Λ, there exists α ∈ K× such that ‖αf‖0 = 1;replacing f by αf , we may and do assume that ‖f‖0 = 1. Since ϕf has a zero, by the above we havethat F = − log |f ϕσ| is monotonically increasing, so F (r) > 0 for all r ∈ (0,∞). For any r ∈ (0,∞),the maximal point of val−1(r) ⊂ B(1)+ is equal to σ(r), so for any y ∈ B(1)+ such that val(y) = r wehave F (r) ≤ − log |f ϕ(y)|. It follows that |f ϕ(y)| < 1 for all y ∈ B(1)+, so ϕ(f(B(1)+)) ⊂ B(1)+.Since B(1)+ is dense in B(1)+∪x, the image of x under ϕf is contained in the closure B(1)+∪xof B(1)+ in B(1). Since 1 = ‖f‖0 = |f ϕ(x)|, the point f ϕ(x) /∈ B(1)+, so f ϕ(x) = x, which is atype-2 point of A1,an. Therefore ϕ(x) is a type-2 point of Xan. n

Applying Lemma 3.3 to a morphism ϕ : B(1)+ → B(1)+ ⊂ A1,an, we see that any automorphismof B(1)+ extends (uniquely) to a homeomorphism B(1)+ ∪ x → B(1)+ ∪ x, so it makes sense tospeak of the end of any open ball. If B is an open ball with end x, we let B denote B ∪ x.

Let B be an open ball with end x. We define a partial ordering on B by declaring y ≤ z if y ∈ [x, z];again see [BR10, §1.4]. Equivalently, for y, z ∈ B we have y ≤ z if and only if |f(y)| ≥ |f(z)| for allanalytic functions f on B. The following lemma is proved as in [BPR11, Proposition 5.27].

Lemma 3.4. Let B be an open ball and let y ∈ B be a type-2 point. Then B \ y is a disjoint union ofthe open annulus A = z ∈ B : z 6≥ y with infinitely many open balls, and By = z ∈ B : z ≥ y isan affinoid subdomain of B isomorphic to the closed ball B(1).

3.5. Open annuli. Let a ∈ mR \ 0. The closure of S(a)+ in B(1) consists of S(a)+ and the twotype-2 points x = σ(0) and y = σ(val(a)), called the ends of S(a)+: again see the proof of [BPR11,Lemma 5.16]. The valuation map val : S(a)+ → (0, val(a)) extends to a continuous map val : S(a)+ ∪x, y → [0, val(a)], and for any z ∈ S(a)+ the geodesic segments [x, z] and [y, z] are contained inS(a)+ ∪ x, y. The following lemma is proved in the same way as the first part of Lemma 3.3.

Lemma 3.6. Let X = Spec(A) be an irreducible affine K-curve, let a ∈ mR \ 0, and let ϕ : S(a)+ →Xan be a morphism with finite fibers. Let x = σ(0) and y = σ(val(a)) be the ends of S(a)+. Then ϕextends in a unique way to a continuous map S(a)+ ∪ x, y → Xan.

It follows from Lemma 3.6 that any automorphism of S(a)+ extends to a homeomorphism S(a)+ ∪x, y ∼−→ S(a)+ ∪ x, y, so it makes sense to speak of the ends of any open annulus. If A is an openannulus with ends x, y, then we let A denote the compact space A ∪ x, y.

The closure of the punctured open ball S(0)+ in B(1) is equal to S(0)+ ∪ 0, x, where x is theend of B(1)+. We define x to be the end of S(0)+ and 0 to be the puncture. As above, the end andpuncture of S(0)+ are isomorphism invariants, so it makes sense to speak of the end and the punctureof any punctured open ball. If A is a punctured open ball with end x and puncture y, then we let Adenote the compact space A ∪ x, y.

LetA be a generalized open annulus. Each connected component ofA\Σ(A) is an open ball [BPR11,Lemma 5.12], and if B is such a connected component with end x, then the inclusion B → A extendsto an inclusion B → A with x mapping into Σ(A); the image of B is the closure of B in A. We definethe retraction to the skeleton τ : A→ Σ(A) by fixing Σ(A) and sending each connected component ofA \Σ(A) to its end. If A = S(a)+, then the retraction τ coincides with σ val : S(a)+ → Σ(S(a)+); inparticular, if (r, s) ⊂ (0, val(a)) = S(a)+, then τ−1((r, s)) = val−1((r, s)).

3.7. The skeleton of a curve. Let X be a smooth, connected, proper algebraic K-curve and letD ⊂ X(K) be a finite set of closed points. The set H(Xan) is natually a metric space [BPR11,Corollary 5.62], although the metric topology on H(Xan) is much finer than the topology inducedby the topology on the K-analytic space Xan. The metric on H(Xan) is locally modeled on an R-tree [BPR11, Proposition 5.63].


Definition 3.8. A semistable vertex set of X is a finite set V of type-2 points of Xan such that Xan \ Vis a disjoint union of open balls and finitely many open annuli. A semistable vertex set of (X,D) isa semistable vertex set of X such that the points of D are contained in distinct open ball connectedcomponents of Xan \ V .

It is a consequence of the semistable reduction theorem of Deligne-Mumford that there existsemistable vertex sets of (X,D): see [BPR11, Theorem 5.49]. In the sequel it will be convenientto consider a curve along with a choice of semistable vertex set, so we give such an object a name.

Definition 3.9. A triangulated punctured curve (X,V ∪D) is a smooth, connected, proper algebraicK-curve X equipped with a finite set D ⊂ X(K) of punctures and a semistable vertex set V of (X,D).

Remark 3.10. This terminology is loosely based on that used in [Duc08] as well as the forthcomingbook of Ducros on analytic spaces and analytic curves. Strictly speaking, what we have defined shouldbe called a semistably triangulated punctured curve, but as these are the only triangulations that weconsider, we will not need the added precision.

3.11. Let (X,V ∪D) be a triangulated punctured curve, so

(3.11.1) Xan \ (V ∪D) = A1 ∪ · · · ∪An ∪⋃α

Bα,

where each Ai is a generalized open annulus and Bα is an infinite collection of open balls. Theskeleton of (X,V ∪D) is the subset

Σ(X,V ∪D) = V ∪D ∪ Σ(A1) ∪ · · · ∪ Σ(An).

(This set is denoted Σ(X,V ∪D) in [BPR11].) For each i and each α the inclusions H(Ai) → H(Xan)and H(Bα) → H(Xan) are local isometries [BPR11, Proposition 5.60]. For each open ball Bα themap Bα → Xan extends to an inclusion Bα → Xan sending the end of Bα to a point xα ∈ V . We saythat Bα is adjacent to xα. For each open annulus Ai the map Ai → Xan extends to a continuous mapAi → Xan sending the ends of Ai to points xi, yi ∈ V . We say that Ai is adjacent to xi and yi. Thelength `(ei) ∈ Λ of the geodesic segment ei = Σ(Ai) ∪ xi, yi is then the modulus of Ai. We say thatV is strongly semistable if xi 6= yi for each open annulus Ai. For each generalized open annulus Ai themap Ai → Xan extends to a continuous map Ai → Xan sending the end of Ai to a point xi ∈ V andsending the puncture to a point yi ∈ D. We say that Ai is adjacent to xi and yi. We define the lengthof ei = Σ(Ai) ∪ xi, yi to be `(ei) =∞.

The skeleton Σ = Σ(X,V ∪D) naturally has the structure of a Λ-metric graph with distinguishedfinite vertex set Vf (Σ) = V , infinite vertex set V∞(Σ) = D, and edges e1, . . . , en as above. Notethat the Λ-points of Σ are exactly the type-2 points of Xan contained in Σ. For x ∈ V the residue fieldH (x) of the completed residue field H (x) at the type-2 point x is a finitely generated field extensionof k of transcendence degree 1; we let Cx be the smooth k-curve with function field H (x). For x ∈ Vwe let g(x) be the genus of Cx, and for x ∈ Σ \ V we set g(x) = 0. These extra data give Σ thestructure of an augmented Λ-metric graph. (In (3.22) we will see that Σ is in fact naturally a metrizedcomplex of k-curves.) By the genus formula [BPR11, (5.45.1)], the genus of X is equal to the genusof the augmented Λ-metric graph Σ.

The open analytic domain Xan \Σ is isomorphic to an infinite disjoint union of open balls [BPR11,Lemma 5.18(3)]. If B is a connected component of Xan \ Σ, then the inclusion B → Xan extends toa map B → Xan sending the end of B to a point of Σ. We define the retraction τ = τΣ : Xan → Σ byfixing Σ and sending a point x ∈ Xan not in Σ to the end of the connected component of x in Xan \Σ.This is a continuous map. If x ∈ Bα is in an open ball connected component of Xan \ (V ∪D), thenτ(x) ∈ V is the end of Bα, and if x ∈ Ai, then τ(x) coincides with the image of x under the retractionmap τ : Ai → Σ(Ai).

Here we collect some additional facts about skeleta from [BPR11, §5].

Proposition 3.12. Let (X,V ∪D) be a triangulated punctured curve with skeleton Σ = Σ(X,V ∪D).


(1) The skeleton Σ is the set of points in Xan that do not admit an open neighborhood isomorphic toB(1)+ and disjoint from V ∪D.

(2) Let V1 be a semistable vertex set of (X,D) such that V1 ⊃ V . Then Σ(X,V1 ∪D) ⊃ Σ(X,V ∪D)and τΣ(X,V ∪D) = τΣ(X,V ∪D) τΣ(X,V1∪D).

(3) Let W ⊂ Xan be a finite set of type-2 points. Then there exists a semistable vertex set of (X,D)containing V ∪W .

(4) Let W ⊂ Σ be a finite set of type-2 points. Then V ∪W is a semistable vertex set of (X,D) andΣ(X,V ∪D) = Σ(X,V ∪W ∪D).

(5) Let x, y ∈ Σ ∩H(Xan). Then any geodesic segment from x to y in H(Xan) is contained in Σ.(6) Let x, y ∈ Xan be points of type-2 or 3 and let [x, y] be a geodesic segment from x to y in H(Xan).

Then there exists a semistable vertex set V1 of (X,D) such that V1 ⊃ V and [x, y] ⊂ Σ(X,V1∪D).

Definition 3.13. A skeleton of (X,D) is a subset of Xan of the form Σ = Σ(X,V ∪ D) for somesemistable vertex set V of (X,D). Such a semistable vertex set V is called a vertex set for Σ. A skeletonof X is a skeleton of (X, ∅).

The augmented Λ-metric graph structure of a skeleton Σ of (X,D) does not depend on the choiceof vertex set for Σ.

3.14. Modifying the skeleton. Let X be a smooth, proper, connected K-curve and let D ⊂ X(K)be a finite set. Let Σ be a skeleton of (X,D), let y ∈ Xan \ Σ, let B be the connected componentof Xan \ Σ containing y, and let x = τ(y) ∈ Σ be the end of B. If y ∈ H(Xan), then the geodesicsegment [x, y] ⊂ B is the unique geodesic segment in H(Xan) connecting x and y. If y ∈ X(K), thenwe define [x, y] to be the geodesic segment [x, y] ⊂ B as in (3.1). The following strengthening ofProposition 3.12(3) will be important in the sequel.

Lemma 3.15. Let V be a semistable vertex set of (X,D), let Σ = Σ(X,V ∪D), let W ⊂ Xan be a finiteset of type-2 points, and let E ⊂ X(K) be a finite set of type-1 points.

(1) There exists a minimal semistable vertex set V1 of (X,D∪E) which contains V ∪W , in the sensethat any other such semistable vertex set contains V1.

(2) Let B be a connected component of Xan \ Σ with end x and let y1, . . . , yn be the points of(W ∪ E) ∩B. Then

Σ(X,V1 ∪D ∪ E) ∩B = [x, y1] ∪ · · · ∪ [x, yn]

if n > 0, and Σ(X,V1 ∪D ∪ E) ∩B = x otherwise. Therefore

Σ(X,V1 ∪D ∪ E) = Σ ∪⋃

y∈W∪E[τΣ(y), y].

(3) The skeleton Σ1 = Σ(X,V1 ∪D ∪ E) is minimal in the sense that any other skeleton containingΣ and W ∪ E must contain Σ1.

Proof. To prove the first part we may assume that W = y consists of a single type-2 point notcontained in Σ and E = ∅, or E = y is a single type-1 point not contained in D and W = ∅. Inthe first case one sees using Lemma 3.4 that V1 = V ∪ y, τ(y) is the minimal semistable vertex setof (X,D) containing V and y, and in the second case V1 = V ∪ τ(y) is the minimal semistablevertex set of (X,D ∪ y) containing V . For W and E arbitrary, let V1 be the minimal semistablevertex set of (X,D ∪ E) containing V ∪ W and let Σ1 = Σ(X,V1 ∪ D). If E = ∅, then it is clearfrom Proposition 3.12(5) that [x, yi] ⊂ Σ1 for each i; the other inclusion is proved by induction onn, adding one point at a time as above. The case E 6= ∅ is similar and is left to the reader. The finalassertion follows easily from the first two and Proposition 3.12. n

Remark 3.16. The above lemma shows in particular that the metric graph Σ(X,V1 ∪ D ∪ E) isobtained from Σ(X,V ∪ D) by a sequence of tropical modifications and their inverses (see Defini-tion II.2.12 and Example II.2.16). It is easy to see that any tropical modification of Σ(X,V ∪ D) is


of the form Σ(X,V1 ∪ D ∪ E) for a semistable vertex set V1 which contains V and a finite subsetE ⊂ X(K).

3.17. The minimal skeleton. The Euler characteristic of X \ D is defined to be χ(X \ D) = 2 −2g(X)−#D, where g(X) is the genus of X. We say that (X,D) is stable if χ(X \D) < 0. A semistablevertex set V of (X,D) is minimal if there is no semistable vertex set of (X,D) properly contained inV , and V is stable if there is no x ∈ V of genus zero (resp. one) and valency less than 3 (resp. one) inΣ(X,V ∪D). It is clear that minimal semistable vertex sets exist. A skeleton Σ of (X,D) is minimal ifthere is no skeleton of (X,D) properly contained in Σ.

The following consequence of the stable reduction theorem can be found in [BPR11, Theorem 5.49].

Proposition 3.18. Let V be a minimal semistable vertex set of (X,D) and let Σ = Σ(X,V ∪D).

(1) If χ(X \D) ≤ 0, then Σ is the unique minimal skleton of (X,D); moreover, Σ is equal to the setof points of Xan \D that do not admit an open neighborhood isomorphic to B(1)+ and disjointfrom D.

(2) If χ(X \D) < 0, then V is the unique minimal semistable vertex set of (X,D), V is stable, and

V = x ∈ Σ : x has valency ≥ 3 or genus ≥ 1.

Remark 3.19. We have χ(X \D) > 0 if and only if X ∼= P1 and D contains at most one point; in thiscase, any type-2 point of Xan serves as a minimal semistable vertex set of (X,D), and there does notexist a unique minimal skeleton. If χ(X \D) = 0, then either X ∼= P1 and D consists of two points,or X is an elliptic curve and D is empty. In the first case, the minimal skeleton Σ of (X,D) is theextended line connecting the points of D and any type-2 point on Σ is a minimal semistable vertexset. If X is an elliptic curve and Xan contains a type-2 point x of genus 1, then x is both the uniqueminimal semistable vertex set and the minimal skeleton. Otherwise X is a Tate curve, Σ is a circle,and any type-2 point of Σ is a minimal semistable vertex set. See [BPR11, Remark 5.51].

3.20. Tangent directions and the slope formula. As above we letX be a smooth, proper, connectedalgebraic K-curve. A continuous function F : Xan → R ∪ ±∞ is called piecewise affine providedthat F (H(Xan)) ⊂ R and F α : [a, b]→ R is a piecewise affine function for every geodesic segmentα : [a, b] → H(Xan). To any point x ∈ Xan is associated a set Tx of tangent directions at x, defined asthe set of germs of geodesic segments in Xan beginning at x. If F : Xan → R ∪ ±∞ is a piecewiseaffine function and v ∈ Tx we denote by dvF (x) the outgoing slope of F in the direction v. We say thatF is harmonic at a point x ∈ Xan provided that there are only finitely many v ∈ Tx with dvF (x) 6= 0,and

∑v∈Tx dvF (x) = 0. See [BPR11, 5.65].

Let x ∈ Xan be a type-2 point, let H (x) be the completed residue field ofXan at x, and let H (x) beits residue field, as in (3.11). Then H (x) is a finitely generated field extension of k of transcendencedegree 1, and there is a canonical bijection between the set Tx of tangent directions to Xan at xand the set DV(H (x)/k) of discrete valuations on H (x) which are trivial on k. For v ∈ Tx we letordv : H (x) → Z denote the corresponding valuation. Let f be a nonzero analytic function on Xan

defined on a neighborhood of x, let c ∈ K× be such that |c| = |f(x)|, and let fx ∈ H (x) denote theresidue of c−1f . Then fx is defined up to multiplication by k×, so for any v ∈ Tx the integer ordv(fx)is well-defined.

The following theorem is called the slope formula in [BPR11, Theorem 5.69]:

Theorem 3.21. Let f ∈ K(X)× be a nonzero rational function on X and let F = − log |f | : Xan →R ∪ ±∞. Let D ⊂ X(K) contain the zeros and poles of f and let Σ be a skeleton of (X,D). Then:

(1) F = F τ , where τ : Xan → Σ is the retraction.(2) F is piecewise affine with integer slopes, and F is affine on each edge of Σ (with respect to a

choice of vertex set V ).(3) If x is a type-2 point of Xan and v ∈ Tx, then dvF (x) = ordv(fx).(4) F is harmonic at all points of Xan \D.


(5) Let x ∈ X(K) and let v be the unique tangent direction at x. Then dvF (x) = ordx(f).

3.22. The skeleton as a metrized complex of curves. Let (X,V ∪D) be a triangulated puncturedcurve with skeleton Σ = Σ(X,V ∪ D). Recall that Σ is an augmented Λ-metric graph with infinitevertices D. We enrich Σ with the structure of a Λ-metrized complex of k-curves as follows. For x ∈ Vlet Cx be the smooth, proper, connected k-curve with function field H (x) as in (3.11). By definitionCx has genus g(x). We have natural bijections

(3.22.1) Tx ∼= DV(H (x)/k) ∼= Cx(k)

where the first bijection v 7→ ordv is defined in (3.20) and the second associates to a closed point ξ ∈Cx(k) the discrete valuation ordξ on the function field H (x) of Cx. Let v ∈ Tx be a tangent directionand define redx(v) to be the point of Cx corresponding to the discrete valuation ordv ∈ DV(H (x)/k).These data make Σ into a Λ-metrized complex of k-curves.

3.23. Lifting metrized complexes of curves. We now prove that every metrized complex of curvesover k arises as the skeleton of a smooth, proper, connectedK-curve. This fact appears in the literaturein various contexts (over discretely-valued fields): see for instance [Bak08, Appendix B] and [Saï96,Lemme 6.3]. For this reason we only sketch a proof using our methods.

Theorem 3.24. Let C be a Λ-metrized complex of k-curves. There exists a triangulated punctured curve(X,V ∪D) such that the skeleton Σ(X,V ∪D) is isomorphic to C.

Proof. Let C be a smooth, proper, connected k-curve. By elementary deformation theory, there isa smooth, proper admissible formal R-scheme C with special fiber C. By GAGA the analytic genericfiber CK is the analytification of a smooth, proper, connected K-curve C . There is a reduction mapred : CK → C from the analytic generic fiber of C to (the set underlying) C, under which the inverseimage of the generic point of C is a single distinguished point x. The set x is a semistable vertexset of C , with associated skeleton also equal to x. Moreover, there is a canonical identification ofH (x) with the field of rational functions on C by [Ber90, Proposition 2.4.4]. See (5.6) for a moredetailed discussion of the relationship between semistable vertex sets and admissible formal models.

Let Γ be the metric graph underlying C. Let V be the vertices of Γ and let E be its edges. Byadding valence-2 vertices we may assume that Γ has no loop edges. Assume for the moment thatΓ has no infinite edges. For a vertex x ∈ V let Cx denote the smooth, proper, connected k-curveassociated to x, and choose an admissible formal curve Cx with special fiber isomorphic to Cx asabove. For clarity we let redCx denote the reduction map (Cx)K → Cx. By [BL93, Proposition 2.2],for every x ∈ Cx(k) the formal fiber red−1

Cx(x) is isomorphic to B(1)+. Let e be an edge of Γ with

endpoints x, y, and let xe ∈ Cx(k), ye ∈ Cy(k) be the reductions of the tangent vectors in the directionof e at x, y, respectively. Remove closed balls from red−1

Cx(xe) and red−1

Cy(ye) whose radii are such

that the remaining open annuli have modulus equal to `(e). We form a new analytic curve Xan bygluing these annuli together using some isomorphism of annuli for each edge e. The resulting curveis proper, hence is the analytification of an algebraic curve X. By construction the image of the set ofdistinguished points of the curves (Cx)K in Xan is a semistable vertex set, and the resulting skeleton(considered as a metric graph) is isomorphic to Γ.

If Γ does have infinite edges, then we apply the above procedure to the finite part of Γ, then punc-ture the curve X in the formal fibers over the smooth points of the residue curves which correspondto the directions of the infinite tails of Γ. n

As an immediate application, we obtain the following property of the “abstract tropicalization map”from the moduli space of stable marked curves to the moduli space of stable abstract tropical curves:

Corollary 3.25. If g and n are nonnegative integers with 2−2g−n < 0, the natural map trop : Mg,n →M tropg,n (see [BPR11, Remark 5.52]) is surjective.

See [ACP12] for an interpretation of the above map as a contraction from Mang,n onto its skeleton

(which also implies surjectivity).


4. MORPHISMS BETWEEN CURVES AND THEIR SKELETA

This section is a relative version of the previous one, in that we propose to study the behavior ofsemistable vertex sets and skeleta under finite morphisms of curves. We introduce finite morphismsof triangulated punctured curves and we prove that any finite morphism of punctured curves can beenriched to a finite morphism of triangulated punctured curves. This powerful result can be usedto prove the simultaneous semistable reduction theorems of Coleman, Liu–Lorenzini, and Liu, whichwe do in Section 5. It is interesting to note that we use only analytic methods on analytic K-curves,making (almost) no explicit reference to semistable models; hence our proofs of the results of Liu–Lorenzini and Liu are very different from theirs. Using a relative version of the slope formula, wealso show that a finite morphism of triangulated punctured curves induces (by restricting to skeleta)a finite harmonic morphism of metrized complexes of curves, which will be crucial in the sequel.

4.1. Morphisms between open balls and generalized open annuli. The main results of this sectionrest on a careful study of the behavior of certain morphisms between open balls and generalized openannuli. Some of the lemmas and intermediate results appear in some form in the literature — see inparticular [Ber90] and [Ber93] — although they are hard to find in the form we need them. As theproofs are not difficult, for the convenience of the reader we include complete arguments.

Lemma 4.2. Let a, a′ ∈ mR and let ϕ : S(a′)+ → S(a)+ be a morphism.

(1) If val ϕ is constant on Σ(S(a′)+), then ϕ(S(a′)+) is contained in S(a)+ \ Σ(S(a)+).(2) If val ϕ is not constant on Σ(S(a′)+), then ϕ(Σ(S(a′)+)) ⊂ Σ(S(a)) and the restriction of ϕ to

Σ(S(a′)+) has the form x 7→ m · x+ b for some nonzero integer m and some b ∈ Λ.(3) If ϕ extends to a continuous map ϕ : S(a′)+ → S(a)+ taking an end of S(a′)+ to an end of

S(a)+, then val ϕ is not constant on Σ(S(a′)+).

Proof. Part (2) is exactly [BPR11, Proposition 5.5], so suppose that val ϕ is constant on Σ(S(a)).The map ϕ : S(a′)+ → S(a)+ ⊂ Gan

m is given by a unit f on S(a′)+. By [BPR11, Proposition 5.2], wecan write f = α (1 + g), where α ∈ K× and |g(x′)| < 1 for all x′ ∈ S(a′)+. Hence |ϕ(x′)−α| < |α| forall x′ ∈ S(a′)+; here |ϕ(x′) − α| should be interpreted as the absolute value of the function f − α inH (x′), or equivalently as the absolute value of the function t−α in H (ϕ(x)), where t is a parameteron Gm. But |t(x)− α| ≥ |α| for every x ∈ Σ(S(a)), so ϕ(S(a′)+) ∩ Σ(S(a)) = ∅.

Let x′ be an end of S(a′)+, let x be an end of S(a)+, and suppose that ϕ(x′) = x. Since x′ isa limit point of Σ(S(a′)+), there exists a sequence of points x′1, x

′2, . . . ∈ Σ(S(a′)+) converging to

x′. Then ϕ(x′1), ϕ(x′2), . . . converge to x, so val(ϕ(x′1)), val(ϕ(x′2)), . . . converges to val(x). Since eachval(ϕ(x′i)) ∈ (0, val(a)) but val(x) /∈ (0, val(a)), val ϕ is not constant on Σ(S(a′)+). n

Lemma 4.3. Let B′ be an open ball, let A be a generalized open annulus, and let ϕ : B′ → A be amorphism. Then ϕ(B′) ∩ Σ(A) = ∅, and ϕ extends to a continuous map B′ → A.

Proof. The assertions of the Lemma are clear if ϕ is constant, so assume that ϕ is non-constant.We identify A with S(a)+ for a ∈ mR and B′ with B(1)+. Any unit on B′ has constant absolute value,so val ϕ is constant on B′. Since B(1)+ \ 0 is a generalized open annulus, and since the type-1point 0 maps to a type-1 point of A, it follows from Lemma 4.2 that ϕ(B′) ∩ Σ(A) = ∅. Let B be theconnected component of A \ Σ(A) containing ϕ(B′). Then the morphism B′ → B → A extends to acontinuous map B′ → B → A. n

Definition 4.4. Let T be a metric space and let m ∈ R>0. A continuous injection ϕ : [a, b] → T is anembedding with expansion factor m provided that r 7→ ϕ(r/m) : [ma,mb] → T is a geodesic segment.A continuous injection ϕ : [a, b] → T is piecewise affine provided that there exist a = a0 < a1 < · · · <ar = b and m1, . . . ,mr ∈ R>0 such that ϕ|[ai−1,ai] is an embedding with expansion factor mi for eachi = 1, . . . , r.

Lemma 4.5. Let B,B′ be open balls, let x′ be the end of B, and let ϕ : B′ → B be a morphism withfinite fibers. Let y′ ∈ B′, let y = ϕ(y′), and let x = ϕ(x′). Then the restriction of ϕ to the geodesic


segment [x′, y′] is injective, and ϕ([x′, y′]) is equal to the geodesic segment [x, y]. If in addition there existsN > 0 such that all fibers of ϕ have fewer than N elements, then ϕ|[x′,y′] is piecewise affine.

Proof. First suppose that y′ does not have type 4. Choose isomorphisms B′ ∼= B(1)+ such that 0 >y′ and B ∼= B(1)+ such that ϕ(0) = 0. Since y′ ∈ [x′, 0] we may replace [x′, y′] by the larger geodesicsegment [x′, 0] to assume that y′ = 0. Define F : Σ(S(0)+) → R by F (x) = val ϕ(x) = − log |ϕ(x)|.By [BPR11, Proposition 5.10], F is a piecewise affine function, and for any point z′ ∈ Σ(S(0)+) atwhich F is differentiable, the derivative of F is equal to the number of zeros q′ of ϕ with val(q′) >val(z′). It follows that F is monotonically increasing, so ϕ is injective on [x′, 0]. Let

Z = τ(q′) : q′ ∈ S(0)+, ϕ(q′) = 0 ⊂ Σ(S(0)+).

If C ⊂ Σ(S(0)+) \ Z is a connected component, then A′ = τ−1(C) is a generalized open annulusmapping to S(0)+. By the above, val ϕ is not constant on Σ(A′) = Σ(S(0)+) ∩ A′, so by Lemma 4.2,ϕ(Σ(A′)) ⊂ Σ(S(0)+). Since Σ(S(0)+) \Z is dense in Σ(S(0)+), we have that [x′, 0] = Σ(S(0)+)∪0maps bijectively onto [x, 0]. It is clear in this case that the restriction of ϕ to the closure of anyconnected component of Σ(S(0)+) \ Z is an embedding with integer expansion factor, so ϕ|[x′,0] ispiecewise affine. Note that ϕ|[x′,0] changes expansion factor at most #ϕ−1(0) times.

Now suppose that y′ has type 4. Suppose that z′, w′ ∈ [x′, y′] are two distinct points such thatϕ(z′) = ϕ(w′). Assume without loss of generality that z′ < w′. Since w′ and z′ have the same imageunder ϕ, they both have the same type, so w′ 6= y′ because y′ is the only type-4 point in [x′, y′].Applying the above to the geodesic [x′, w′] gives a contradiction. Therefore, ϕ is injective on [x′, y′],so ϕ([x′, y′]) = [x, y] since B is uniquely path-connected ([BR10, Corollary 1.14]).

Assume now that all fibers of ϕ have size bounded by N . The above argument proves that for allz′ < y′, the restriction of ϕ to [x′, z′] is piecewise affine, and ϕ|[x′,z′] changes expansion factor at mostN times. Therefore there exists z′0 < y′ and m ∈ Z>0 such that for all z′ ∈ [z′0, y

′], the restriction ofϕ to [z′0, z

′] is an embedding with expansion factor m. It follows that ϕ|[z′0,y′] is an embedding withexpansion factor m, so ϕ|[x′,y′] is piecewise affine. n

Lemma 4.6. Let B,B′ be open balls and let ϕ : B′ → B be a morphism. Suppose that ϕ is open,separated, and has finite fibers, so it extends canonically to a continuous map ϕ : B′ → B by Lemma 3.3.Let x′ be the end of B′ and let x = ϕ(x′) ∈ B. Then B1 = ϕ(B′) is an open ball connected componentof B \ x and ϕ : B′ → B1 is finite and order-preserving. In particular, if x is the end of B, thenϕ : B′ → B is finite and order-preserving.

Proof. Let y′ ∈ B′. By Lemma 4.5, the restriction of ϕ to the geodesic [x′, y′] is injective, soϕ(y′) 6= x. Therefore x /∈ B1. Let C be the connected component of B \ x containing B1. Since x isan end of C and x′ maps to x, Lemma 4.3 implies that C is not an open annulus, so it is an open ball.By hypothesis, B1 is an open subset of C; since B′∪x′ is compact, ϕ(B′∪x′) = B1∪x is closedin B, and therefore B1 = (B1 ∪ x) ∩ C is closed in C. Since C is connected, we have B1 = C.

Since ϕ has finite fibers, ϕ : B′ → B1 is finite if and only if it is proper by [Ber90, Corollary 3.3.8].If D ⊂ B1 is compact, then ϕ−1(D) is compact since ϕ−1(D) is closed as a subset of the compact spaceB′. One has Int(B′/B1) = B′ by [Ber90, Proposition 3.1.3(i)] since B′ is a boundaryless K-analyticspace. Therefore ϕ is proper.

The fact that ϕ : B′ → B1 is order-preserving follows immediately from Lemma 4.5. n

4.7. Let ϕ : B′ → B be a finite morphism of open balls. Since an open ball is a smooth curve, ϕ is flatin the sense that if M (A) ⊂ B is an affinoid domain and ϕ−1(M (A)) = M (A′), then A′ is a (finite)flat A-algebra. For x ∈M (A) the fiber over x is

ϕ−1(x) = M(A′⊗AH (x)

)= M

(A′ ⊗AH (x)

),

where the second equality holds because A′ is a finite A-algebra. It follows that for x ∈ B the quantity∑x′ 7→x dimH (x) Oϕ−1(x),x′ is independent of x; we call this number the degree of ϕ.

Proposition 4.8. Let B,B′ be open balls and let ϕ : B′ → B be a finite morphism. Let x′ be the endof B′, let x = ϕ(x′) be the end of B, let y ∈ B, and let y′1, . . . , y

′n ∈ B′ be the inverse images of y in B′.


Thenϕ−1([x, y]) = [x′, y′1] ∪ · · · ∪ [x′, y′n].

Proof. Let T ′ = [x′, y′1] ∪ · · · ∪ [x′, y′n]. The inclusion T ′ ⊂ ϕ−1([x, y]) follows from Lemma 4.5.First we claim that for z ∈ [x, y] near enough to x, there is only one preimage of z in B′. Shrinking[x, y] if necessary, we may assume that y has type 2. Choose a type-1 point w ∈ B such that w > y,so (x, y] ⊂ (x,w) = Σ(B \ w). Let w′ ∈ B′ be a preimage of w and choose z′ ∈ Σ(B′ \ w′) suchthat z′ ≤ τ(q′) for all q′ ∈ ϕ−1(w), where τ : B′ \ w′ → Σ(B′ \ w′) is the retraction. Let A′ be theopen annulus τ−1((x′, z′)) ⊂ B′. Then ϕ(A′) ⊂ B \ w, and ϕ takes the end x′ of A′ to the end x ofB \w, so by Lemma 4.2(2,3), ϕ(Σ(A′)) ⊂ Σ(B \w) and the map Σ(A′)→ Σ(B \w) is injective.If u′ ∈ A′ \ Σ(A′), then u′ is contained in an open ball in A′, so ϕ(u′) /∈ Σ(B \ w) by Lemma 4.3.Therefore every point of Σ(B \ w) has at most one preimage in A′. Let u′ ∈ B′ \ A′, so u′ ≥ z′.Then ϕ(u′) ≥ ϕ(z′), so every point u ∈ (x, y] with u < ϕ(z′) has exactly one preimage in B′ (note thatϕ(z′) ∈ Σ(B \ w) by Lemma 4.5).

Let d be the degree of the finite morphism ϕ, so for every z ∈ B we have

d =∑

z′∈ϕ−1(z)

dimH (z) Oϕ−1(z),z′ .

For y ∈ B as in the statement of the Proposition, define a function δ : (x, y]→ Z by

δ(z) =∑

z′∈ϕ−1(z)z′∈T ′

dimH (z) Oϕ−1(z),z′ .

Clearly δ(z) ≤ d for all z ∈ (x, y], and δ(z) = d if and only if ϕ−1(z) ⊂ T ′. By definition of T ′ we haveδ(y) = d, and by the above, δ(z) = d for z ∈ (x, y] close enough to x (any geodesic segment [x′, y′i]surjects onto [x, y], hence contains the unique preimage of z). Therefore it is enough to show thatδ(z1) ≥ δ(z2) if z1 ≤ z2.

If z ∈ B is a point of type 2 or 3, then Bz B w ∈ B : w ≥ z is a (not necessarily strict)affinoid subdomain of B. If w′ ∈ B′ is such that ϕ(w′) ≥ z, then ϕ([x′, w′]) = [x, ϕ(w′)] is a geodesiccontaining z, so there exists z′ ∈ [x′, w′] mapping to z. It follows that

ϕ−1(Bz) =∐z′ 7→zw′ ∈ B′ : w′ ≥ z′ =

∐z′ 7→z

B′z′

is a disjoint union of affinoid domains. Hence each map Bz′ → Bz is finite, and its degree is equal todimH (z) Oϕ−1(z),z′ . Let z1, z2 ∈ (x, y] and assume that z1 < z2, so z1 has type 2 or 3. For z′1 ∈ ϕ−1(z1)we have

dimH (z1) Oϕ−1(z1),z′1=∑z′2 7→z2z′2≥z

′1

dimH (z2) Oϕ−1(z2),z′2.

Summing over all z′1 ∈ ϕ−1(z1) ∩ T ′, we obtain

δ(z1) =∑z′1 7→z1z′1∈T

′

dimH (z1) Oϕ−1(z1),z′1=∑z′1 7→z1z′1∈T

′

∑z′2 7→z2z′2≥z

′1

dimH (z2) Oϕ−1(z2),z′2

≥∑z′1 7→z1z′1∈T

′

∑z′2 7→z2z′2≥z

′1

z′2∈T′

dimH (z2) Oϕ−1(z2),z′2= δ(z2),

where the final equality holds because if z′2 ∈ ϕ−1(z2) ∩ T ′, then there exists z′1 ∈ ϕ−1(z1) ∩ T ′ suchthat z′2 ≥ z′1, namely, the unique point of [x′, z′2] mapping to z1. n

Proposition 4.9. Let a ∈ mR \ 0, let A′ = S(a)+, let B be an open ball, and let ϕ : A′ → B bea morphism with finite fibers. Suppose that each end of A′ maps to the end of B or to a type-2 point


of B under the induced map A′ → B. Let α = ϕ σ : [0, val(a)] → B. Then there exist finitely manynumbers r0, r1, r2, . . . , rn ∈ Λ with 0 = r0 < r1 < r2 < · · · < rn = val(a) such that α is an embeddingwith nonzero integer expansion factor when restricted to each interval [ri, ri+1]. In other words, α ispiecewise affine with integer expansion factors. Moreover, the image of α is a geodesic segment betweentype-2 points of B.

Proof. Let x′ be the end σ(0) of A′ and let x = ϕ(x′). Suppose first that x is the end of B. Choosean identification B ∼= B(1)+, and let r+ = minval(y′) : ϕ(y′) = 0 > 0. Let A′+ be the open annulusval−1((0, r+)) ⊂ A′. Then ϕ(A′+) ⊂ B(1)+ \ 0 = S(0)+ and the end x′ of A′+ maps to the end xof S(0)+, so by Lemma 4.2(2,3), α is an embedding with (nonzero) integer expansion factor whenrestricted to [0, r+].

Now suppose that x is not the end of B. Let r = minval(y′) : ϕ(y′) = x > 0 and let A′′ =val−1((0, r)). Then ϕ(A′′) is contained in a connected component C of B \ x, and ϕ takes the endx′ of A′′ to the end x of C. If C is an open annulus, then α is an embedding with integer expansionfactor when restricted to [0, r] by Lemma 4.2(2,3), and if C is an open ball, then we proceed as aboveto find r+ ∈ (0, r) ∩ Λ such that α is an embedding with integer expansion factor when restricted to[0, r+].

Applying the above argument to the morphism ϕ composed with the automorphism t 7→ a/t ofS(a)+ (which interchanges the two ends), we find that there exists r− ∈ [0, val(a)) ∩ Λ such that αis an embedding with integer expansion factor when restricted to [r−, val(a)]. Let s ∈ (0, val(a)) ∩Λ. Replacing A′ with the annulus val−1((0, s)) (resp. val−1((s, val(a)))), the above arguments thenprovide us with s+ ∈ (s, val(a)] ∩ Λ (resp. s− ∈ [0, s) ∩ Λ) such that α is an embedding with integerexpansion factor when restricted to [s, s+] (resp. [s−, s]). The first assertions now follow because thereis a finite subcover of the open covering

[0, r+) ∪ (r−, val(a)] ∪ (s−, s+) : s ∈ (0, r) ∩ Λ

of the compact space [0, r].As for the final assertion, choose 0 = r0 < r1 < r2 < · · · < rn = val(a) such that α is an embedding

with integer expansion factor on each [ri, ri+1]. Let i0 ∈ 0, 1, . . . , n be the largest integer such thatα(ri0) > α(ri) (in the canonical partial ordering on B) for all i < i0. If i0 = n, then we are done,so assume that i0 < n. Let y = α(ri0) ∈ B, and choose an identification B ∼= B(1)+ such that0 > y. If F = − log |ϕ|, then by [BPR11, Proposition 5.10], at every point r ∈ (0, val(a)) the changein slope of F at r is equal to the negative of the number of zeros of ϕ with valuation r (cf. the proofof Lemma 4.5); in particular, the slope of F can only decrease. By construction F is monotonicallyincreasing on [0, ri0 ]. Since α([ri0 , ri0+1]) is a geodesic segment, it meets y only at α(ri0). The imageof (ri0 , ri0+1] under α is not contained in an open ball connected component of B \ y becauseα(ri0+1) 6> y; hence F is decreasing on an interval [ri0 , ri0 + ε] for some ε > 0. Since the slope of Fcan only decrease, it follows that F is monotonically decreasing on [ri0 , rn]. It follows immediatelyfrom this that α([0, val(a)]) = [x, y] or α([0, val(a)]) = [α(val(a)), y], whichever segment is larger. n

4.10. Morphisms between curves and skeleta. In what follows we fix smooth, connected, properalgebraic K-curves X,X ′ and a finite morphism ϕ : X ′ → X. Let D ⊂ X(K) and D′ ⊂ X ′(K) befinite sets of closed points. The map on analytifications ϕ : X ′an → Xan is finite and open by [Ber90,Lemma 3.2.4].

Proposition 4.11. Let Σ′ be a skeleton of (X ′, D′) and let Σ be a skeleton of (X,D). There exists askeleton Σ1 of (X,D ∪ ϕ(D′)) containing Σ ∪ ϕ(Σ′), and there is a minimal such Σ1 with respect toinclusion.

Proof. First we will prove the Proposition in the case D = D′ = ∅. Let V be a vertex set for Σ andlet V ′ be a vertex set for Σ′ containing τ(y′) : y′ ∈ ϕ−1(V ). Let A′ be an open annulus connectedcomponent of X ′an \ V ′ and let e′ ⊂ Σ′ be the associated edge. We claim that ϕ(e′) is a geodesicsegment between type-2 points of Xan. Let C be the connected component of Xan \ V containingϕ(A′).


(i) If C is an open ball then the claim follows immediately from Proposition 4.9.(ii) If C is an open annulus and ϕ(A′)∩Σ(C) = ∅, then A′ is contained in an open ball in C because

each connected component of C \ Σ(C) is an open ball, so the claim is true as in (i).(iii) If C is an open annulus and ϕ(A′) ∩ Σ(C) 6= ∅, then ϕ(e′) is a geodesic segment in Xan by

Lemma 4.2.Applying the above to each edge e′ of Σ′, we find that there exists a finite set of type-2 points

x1, y1, x2, y2, . . . , xn, yn ∈ Xan such that ϕ(Σ′) =⋃ni=1[xi, yi], where [xi, yi] denotes a geodesic seg-

ment from xi to yi. Let

Σ1 = Σ ∪n⋃i=1

([xi, τΣ(xi)] ∪ [yi, τΣ(yi)]

).

By Lemma 3.15(2) as applied to W = x1, y1, . . . , xn, yn, we have that Σ1 is a skeleton of X, so byProposition 3.12(5), Σ1 contains Σ∪ϕ(Σ′) since Σ1 contains each geodesic segment [xi, yi]. Any otherskeleton of X containing Σ and all xi, yi also contains the geodesics [xi, τΣ(xi)] and [yi, τΣ(yi)], so Σ1

is the minimal such skeleton.Now assume that D qD′ 6= ∅. By Lemma 3.15(3) there is a minimal skeleton Σ1 of X containing

Σ and ϕ(D′); replacing Σ by Σ1 and D by D ∪ ϕ(D′), we may assume without loss of generality thatϕ(D′) ⊂ D. We will proceed by induction on the size of D qD′. As above we let V be a vertex set forΣ and we let V ′ be a vertex set for Σ′ containing τ(y′) : y′ ∈ ϕ−1(V ∪D).

(i) Suppose that D′ is not empty. Let x′ ∈ D′ and let Σ1 be the minimal skeleton of (X,D)containing Σ∪ϕ(Σ(X ′, V ′ ∪D′ \ x′)). Let A′ be the connected component of X ′an \ (V ′ ∪D′)whose closure contains x′, let x = ϕ(x′) ∈ D, and let A be the connected component ofXan \ (V ∪D) whose closure contains x. Note that Σ′ = Σ(X ′, V ′ ∪D′ \ x′) ∪ Σ(A′) ∪ x′.Clearly ϕ(A′) ⊂ A, and ϕ−1(x) ∩ A′ = ∅ by construction. Since ϕ takes the puncture of A′ tothe puncture of A, one shows as in the proof of Lemma 4.2 that ϕ(Σ(A′)) ⊂ Σ(A). ThereforeΣ1 contains Σ ∪ ϕ(Σ′).

(ii) Now suppose that D′ = ∅ and D 6= ∅. Let x ∈ D and let Σ1 be the minimal skeleton of(X,D \ x) containing Σ(X,V ∪ D \ x) ∪ ϕ(Σ′). Let Σ2 = Σ1 ∪ [x, τ(x)]. It follows fromLemma 3.15(3) that Σ2 is the minimal skeleton of (X,D) containing Σ1.

n

Let V be a semistable vertex set of (X,D). Recall from (3.11) that a connected component Cof Xan \ (V ∪ D) is adjacent to a vertex x ∈ V provided that the closure of C in Xan contains x.Proposition 4.12(1) below is exactly [Ber90, Theorem 4.5.3] when D = D′ = ∅.Proposition 4.12. Suppose that D′ = ϕ−1(D) and that one of the following two conditions holds:

(1) χ(X \D) ≤ 0 (hence also χ(X ′ \D′) ≤ 0), or(2) ϕ−1(V ) ⊂ V ′.

In the situation of (1) let Σ (resp. Σ′) be the minimal skeleton of (X,D) (resp. (X ′, D′)), and in (2) letΣ = Σ(X,V ∪D) and Σ′ = Σ(X ′, V ′ ∪D′). Then ϕ−1(Σ) ⊂ Σ′.

Proof. First suppose that (2) holds. Let x′ ∈ X ′an \ Σ′ and let B′ be the connected componentof X ′an \ Σ′ containing x′, so B′ is an open ball. By hypothesis, ϕ(B′) is contained in a connectedcomponent C ofXan\(V ∪D). If C is an open ball, then ϕ(B′)∩Σ = ∅ by the definition of Σ(X,V ∪D).If C is a generalized open annulus, then ϕ(B′) ∩ Σ = ϕ(B′) ∩ Σ(C) = ∅ by Lemma 4.3. Thereforeϕ(x′) /∈ Σ.

Now suppose that (1) holds. Let V be a semistable vertex set of (X,D) such that Σ = Σ(X,V ∪D).By subdividing edges of Σ and enlarging V if necessary, we may and do assume that Σ has no loopedges. By Proposition 3.18(2), no point in V of genus zero has valence one in Σ. First we claim thatif B′ ⊂ X ′an \ D′ is an open analytic domain which is isomorphic to B(1)+, then ϕ(B′) ∩ V = ∅. Ifϕ−1(V ) ∩ B′ contains more than one point, then it easy to see that there exists a smaller open ballB′′ ⊂ B′ such that ϕ−1(V )∩B′′ contains exactly one point. Replacing B′ by B′′, we may assume thatthere is a unique point y′ ∈ ϕ−1(V ) ∩B′. Let y = ϕ(y′) ∈ V . Since g(y′) = 0 we have g(y) = 0.


The open analytic domain B′ \ y′ is the disjoint union of an open annulus A′ and an infinitecollection of open balls by Lemma 3.4. Each connected component C ′ of B′ \ y′ maps into aconnected component C of Xan \ (V ∪D) adjacent to y, with the end y′ of C ′ mapping to the end yof C. By Lemma 4.3, no open ball connected component of B′ \ y′ can map to a generalized openannulus connected component of Xan \ (V ∪ D). There are at least two generalized open annulusconnected components A of Xan \ (V ∪D) adjacent to y, so some such A must satisfy ϕ(B′) ∩A = ∅.But the map ϕ : X ′an → Xan is open by [Ber90, Lemma 3.2.4], so ϕ(B′) is an open neighborhood ofy, which contradicts the fact that y is a limit point of A. This proves the claim.

Let V ′ be a semistable vertex set of (X ′, D′) such that Σ′ = Σ(X ′, V ′ ∪ D′). Since X ′ \ Σ′ is adisjoint union of open balls, by the above we have ϕ−1(V ) ⊂ Σ′. Hence we may enlarge V ′ to containϕ−1(V ) without changing Σ′, so we are reduced to (2). n

Theorem 4.13. Let Σ′ be a skeleton of (X ′, D′) and let Σ be a skeleton of (X,D). Suppose that ϕ(Σ′) ⊂Σ, and if X ∼= P1 assume in addition that there exists a type-2 point z ∈ Σ such that ϕ−1(z) ⊂ Σ′. Thenϕ−1(Σ) is a skeleton of (X ′, ϕ−1(D)).

We will require the following lemmas. Lemma 4.14 is similar to [Ber90, Corollary 4.5.4].

Lemma 4.14. Let B′ ⊂ X ′an be an open analytic domain isomorphic to B(1)+ and let B = ϕ(B′) ⊂Xan. Then B is an open analytic domain of Xan, and one of the following is true:

(1) B is an open ball and ϕ : B′ → B is finite and order-preserving.(2) X ∼= P1 and B = Xan.

Proof. Suppose that the genus ofX is at least one, soX ′ also has genus at least one. By Lemma 4.6we only need to show that B is contained in an open ball in Xan. Let Σ (resp. Σ′) be the minimalskeleton of X (resp. X ′). By Proposition 3.18(1) we have B′ ⊂ X ′an \ Σ′, so by Proposition 4.12(1)we have B ⊂ Xan \ Σ. But every connected component of Xan \ Σ is an open ball, so B is containedin an open ball.

In the case X = P1, suppose first that B′(K) → P1(K) is not surjective, so we may assume that∞ /∈ B after choosing a suitable coordinate on P1. Let x′ be the end of B′ and let x = ϕ(x′). Then xis a type-2 point, so ϕ(B′ ∪x′) = B ∪x is a compact subset of A1,an. Since A1,an is covered by anincreasing union of open balls, B is contained in an open ball.

Now suppose thatB′(K)→ P1(K) is surjective. Since P1(K) is dense in P1,an andB∪x is closedin P1,an, it follows that B ∪ x = P1,an. If x /∈ B, then B is contained in a connected component ofP1,an \ x, which contradicts the surjectivity of B′(K)→ P1(K). Therefore B = P1,an. n

Remark 4.15. Case (2) of Lemma 4.14 does occur. For instance, let ϕ : P1 → P1 be the finitemorphism t 7→ t2, and let B′ be the open ball in P1,an obtained by deleting the closed ball of radius1/2 around 1 ∈ P1(K). If char(k) 6= 2, then for every point x ∈ P1(K), either x ∈ B′(K) or−x ∈ B′(K), so B′ → P1,an is surjective.

Lemma 4.16. Let V be a semistable vertex set of (X,D), let Σ = Σ(X,V ∪D), and let B ⊂ X be anopen analytic domain isomorphic to B(1)+ whose end x is contained in Σ. Then

Σ ∩B =⋃

y∈B∩(V ∪D)

[x, y].

Proof. We may assume without loss of generality that x ∈ V . Let V0 = V \ (V ∩ B) and D0 =D \ (D ∩ B). Then V0 is a semistable vertex set of (X,D0): indeed, every connected component ofXan\(V ∪D) is either a connected component ofXan\(V0∪D0) or is contained in B, soXan\(V0∪D0)is a disjoint union of open balls and finitely many generalized open annuli. Let Σ0 = Σ(X,V0∪D0), sox ∈ Σ0 butB∩Σ0 = ∅ by Proposition 3.12(1). By Lemma 4.6 we have thatB is a connected componentof Xan \ Σ0, so the Lemma now follows from Lemma 3.15 as applied to Σ = Σ0, W = V ∩ B, andE = D ∩B. n


Proof of Theorem 4.13. Note that ϕ(Σ′) ⊂ Σ implies ϕ(D′) ⊂ D. Let V (resp. V ′) be a vertex setfor Σ (resp. Σ′). We claim that

ϕ−1(Σ) = Σ′ ∪⋃

x′∈ϕ−1(V ∪D)

[x′, τΣ′(x′)],

which by Lemma 3.15 is the minimal skeleton of (X ′, ϕ−1(D)) containing Σ′ and ϕ−1(V ).Let B′ be a connected component of X ′an \ Σ′, let x′ ∈ Σ′ be its end, and let x = ϕ(x′) ∈ Σ. Then

B = ϕ(B′) is an open ball and x is its end: if X 6∼= P1, then this follows directly from Lemma 4.14,and if X ∼= P1, then B ⊂ Xan \ z, so ϕ(B) 6= Xan and therefore B is an open ball in this case aswell. Hence

Σ ∩B =⋃

y∈B∩(V ∪D)

[x, y]

by Lemma 4.16, so

ϕ−1(Σ) ∩B′ =⋃

y′∈B′∩ϕ−1(V ∪D)

[x′, y′]

by Proposition 4.8. n

Remark 4.17. When X ∼= P1, the extra hypothesis on Σ′ in Theorem 4.13 is necessary. Indeed, letϕ : P1 → P1 be a finite morphism and let x′ ∈ P1,an be a type-2 point such that ϕ−1(ϕ(x′)) has morethan one element. Then Σ = ϕ(x′) is a skeleton containing the image of the skeleton Σ′ = x′, butϕ−1(Σ) is not a skeleton.

Corollary 4.18. Let Σ (resp. Σ′) be a skeleton of (X,D) (resp. (X ′, D′)). There exists a skeleton Σ1 of(X,D ∪ ϕ(D′)) such that Σ1 ⊃ Σ∪ ϕ(Σ′) and such that ϕ−1(Σ1) is a skeleton of (X ′, ϕ−1(D ∪ ϕ(D′))).Moreover, there is a minimal such Σ1 with respect to inclusion.

Proof. If X 6∼= P1 then this is an immediate consequence of Proposition 4.11 and Theorem 4.13:if Σ1 is the minimal skeleton of (X,D ∪ ϕ(D′)) containing Σ ∪ ϕ(Σ′), then ϕ−1(Σ1) is a skeleton of(X ′, ϕ−1(D∪ϕ(D′))). Suppose then that X ∼= P1. Let x′ ∈ Σ′ be a type-2 point, let x = ϕ(x′), and letx′, x′1, x

′2, . . . , x

′n be the points of X ′ mapping to x. Let

Σ′1 = Σ′ ∪n⋃i=1

[x′i, τΣ′(x′i)].

This is the minimal skeleton of (X ′, D′) containing Σ′ and ϕ−1(x) by Lemma 3.15(3). Let Σ1 be theminimal skeleton of (X,D∪ϕ(D′)) containing Σ∪ϕ(Σ′1). Then ϕ−1(Σ1) is a skeleton of (X ′, ϕ−1(D∪ϕ(D′))) by Theorem 4.13. If Σ2 is a skeleton of (X,D∪ϕ(D′)) such that Σ2 ⊃ Σ∪ϕ(Σ′) and ϕ−1(Σ2)is a skeleton of (X ′, ϕ−1(D ∪ ϕ(D′))), then ϕ−1(Σ2) contains each geodesic [x′i, τΣ′(x

′i)], so ϕ−1(Σ2)

contains Σ′1 and therefore Σ2 ⊃ Σ1. n

Remark 4.19. Suppose that ϕ : X ′ → X is a finite morphism such that ϕ−1(D) = D′ and ϕ−1(V ) =V ′. Let Σ = Σ(X,V ∪ D) and Σ′ = Σ(X ′, V ′ ∪ D′). In this case we can describe the skeleton Σ1

of Corollary 4.18 more explicitly, as follows. Let e′ be an open edge of Σ′ with respect to the givenchoice of vertex set and let A′ = τ−1(e′). This is a connected component of X ′an \ (V ′ ∪ D′). Let Abe the connected component of Xan \ (V ∪D) containing ϕ(A′). Since A′ is a connected componentof ϕ−1(A), the map A′ → A is finite, hence surjective. If A′ is a punctured open ball, then so is A, soby Lemma 4.2, ϕ maps e′ homeomorphically with nonzero integer expansion factor onto e = Σ(A). IfA′ is an open annulus, then A could be an open annulus or an open ball; in the former case we againhave e′ mapping homeomorphically onto e = Σ(A) with nonzero integer expansion factor. If A is anopen ball, then by Proposition 4.9, e′ maps onto a geodesic segment in A; clearly both vertices of e′

map to the end of A, so e′ “goes straight and doubles back” under ϕ.It follows from this and Proposition 4.12(2) that ϕ(Σ′) is equal to Σ union a finite number of

geodesic segments T1, . . . , Tr contained in open balls B1, . . . , Br ⊂ Xan \ (V ∪ D) and with one


endpoint at elements of V . These are precisely the images of the edges of Σ′ not mapping to edges ofΣ. By Lemma 3.15, Σ ∪ (T1 ∪ · · · ∪ Tr) is again a skeleton of X, so in this case

Σ1 = ϕ(Σ′) = Σ ∪ (T1 ∪ · · · ∪ Tr)

(the inverse image of Σ1 is a skeleton by Theorem 4.13).

4.20. Tangent directions and morphisms of curves. Our next goal is to prove that a finite morphismof curves induces a finite harmonic morphism of metrized complexes of curves for a suitable choiceof skeleta. First we need to formulate and prove a relative version of Theorem 3.21. Let X,X ′ besmooth, proper, connected K-curves.

Definition 4.21. A continuous function ϕ : X ′an → Xan is piecewise affine provided that, for everygeodesic segment α : [a, b] → X ′an, the composition ϕ α is piecewise affine with integer expansionfactors in the sense of Definition 4.4.

Let ϕ : X ′an → Xan be a piecewise affine function, let x′ ∈ X ′an, and let x = ϕ(x′) ∈ Xan. Letv′ ∈ Tx′ and let α : [a, b] → X ′an be a geodesic segment representing v′ (so a = −∞ if x′ has type 1).Taking b small enough, we can assume that ϕ α is an embedding with integer expansion factor m.Let v ∈ Tx be the tangent direction represented by ϕ α. We define

dϕ(x′) : Tx′ → Tx by dϕ(x′)(v′) = v;

this is independent of the choice of α. We call the expansion factor m the outgoing slope of ϕ in thedirection v′ and we write m = dv′ϕ(x′).

Definition 4.22. A piecewise affine function ϕ : X ′an → Xan is harmonic at a point x′ ∈ X ′an

provided that, for all v ∈ Tϕ(x′), the integer ∑v′∈Tx′dϕ(v′)=v

dv′ϕ(x′)

is independent of the choice of v.

Let ϕ : X ′ → X be a finite morphism of smooth, proper, connected K-curves, let x′ ∈ X ′an be atype-2 point, and let x = ϕ(x′). Let Cx and Cx′ denote the smooth proper connected k-curves withfunction fields H (x) and H (x′), respectively. We denote by ϕx′ the induced morphism Cx′ → Cx.We have the following relative version of the slope formula of Theorem 3.21:

Theorem 4.23. Let ϕ : X ′ → X be a finite morphism of smooth, proper, connected K-curves.

(1) The analytification ϕ : X ′an → Xan is piecewise affine and harmonic.(2) Let x′ ∈ X ′an be a type-2 point, let x = ϕ(x′), let v′ ∈ Tx′ , let v = dϕ(v′) ∈ Tx, and let ξv′ ∈ Cx′

and ξv ∈ Cx be the closed points associated (3.22.1) to v′ and v, respectively. Then ϕx′(ξv′) = ξv,and the ramification degree of ϕx′ at ξv′ is equal to dv′ϕ(x′).

(3) Let x′ ∈ X ′an be a type-1 point and let v′ ∈ Tx′ be the unique tangent direction. Then dv′ϕ(x′)is the ramification degree of ϕ at x′.

Proof. First we claim that ϕ is piecewise affine. Let [x′, y′] ⊂ X ′an be a geodesic segment. Supposefirst that x′ and y′ have type 2 or 3. Then there exists a skeleton Σ′ of X ′ containing [x′, y′] by [BPR11,Corollaries 5.56 and 5.64]. From this we easily reduce to the case that [x′, y′] is an edge of Σ′ withrespect to some vertex set; now the claim follows from Lemma 4.2 and Proposition 4.9 as in the proofof Proposition 4.11. If x′ has type 1 or 4, then there is an open neighborhood B of x = ϕ(x′) and anopen neighborhood B′ ⊂ ϕ−1(B) of x′ such that B and B′ are open balls. Shrinking B′ if necessarywe can assume that y′ /∈ B′, so the end z′ of B′ is contained in [x′, y′] (removing z′ disconnects X ′an).The restriction of ϕ to [x′, z′] is piecewise affine by Lemma 4.5, so this proves the claim.

Next we prove (2). By functoriality of the reduction map, for any non-zero rational function f onX, we have ϕ∗x′(fx) = ϕ∗(f)x′ . By the slope formula (Theorem 3.21), the point ξv corresponds to the


discrete valuation on H (x) given by ordξv (fx) = dvF (x), where F = − log |f |. Similarly, the slope

formula applied to ϕ∗(f) on X ′ gives ordξv′ (ϕ∗(f)x′) = dv′ϕ(x′)dvF (x). Therefore, we get

ordξv′ (ϕ∗x′(fx)) = dv′ϕ(x′) ordξv (fx).

Since any non-zero element of H (x) is of the form fx for some non-zero rational function f on X,

this shows that the center of the valuation ξv′ on H (x)ϕ∗x′

−→ H (x′) coincides with the center of ξv,that is ϕx′(ξv′) = ξv, and at the same time, the ramification degree of ϕx′ at ξv′ is equal to dv′ϕ(x′).

The proof of (3) proceeds in the same way as (2), applying the slope formula to a non-zero rationalfunction f on X with a zero at x = ϕ(x′), and to ϕ∗(f) on X ′. n

4.24. Morphisms of curves induce morphisms of metrized complexes. More precisely, this occurswhen the morphism of curves respects a choice of triangulations, in the following sense.

Definition 4.25. Let (X,V ∪D) and (X ′, V ′∪D′) be triangulated punctured curves. A finite morphismϕ : (X ′, V ′∪D′)→ (X,V ∪D) is a finite morphism ϕ : X ′ → X such that ϕ−1(D) = D′, ϕ−1(V ) = V ′,and ϕ−1(Σ(X,V ∪D)) = Σ(X ′, V ′ ∪D′) (as sets).

Corollary 4.26. Let ϕ : X ′ → X be a finite morphism of smooth, connected, proper K-curves, letD ⊂ X(K) be a finite set, and let D′ = ϕ−1(D). There exists a strongly semistable vertex set V of (X,D)such that V ′ = ϕ−1(V ) is a strongly semistable vertex set of (X ′, D′) and such that

Σ(X ′, V ′ ∪D′) = ϕ−1(Σ(X,V ∪D)).

In particular, ϕ extends to a finite morphism of triangulated punctured curves.Proof. By Corollary 4.18, there is a skeleton Σ of (X,D) such that Σ′ = ϕ−1(Σ) is a skeleton of

(X ′, D′). Let V ′0 be a strongly semistable vertex set of Σ′, let V be a strongly semistable vertex set ofΣ containing ϕ(V ′0), and let V ′ = ϕ−1(V ). Then V ′ is again a strongly semistable vertex set of Σ′. n

4.27. Let ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D) be a finite morphism of triangulated punctured curves.Let Σ = Σ(X,V ∪ D) and Σ′ = Σ(X ′, V ′ ∪ D′). Let e′ be an open edge of Σ′ and let e = ϕ(e′), anopen edge of Σ. The image of the annulus A′ = τ−1(e′) is contained in A = τ−1(e), so the restrictionof ϕ to e′ is an embedding with integer expansion factor by Lemma 4.2(2). It is clear from this andTheorem 4.23 that ϕ|Σ′ : Σ′ → Σ is a harmonic (V ′, V )-morphism of Λ-metric graphs. For x′ ∈ V ′with image x = ϕ(x′) we have a finite morphism of k-curves ϕx′ : Cx′ → Cx. It now follows fromTheorem 4.23 that these extra data enrich ϕ|Σ′ : Σ′ → Σ with the structure of a finite harmonicmorphism of Λ-metrized complexes of curves:

Corollary 4.28. Let ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D) be a finite morphism of triangulated puncturedcurves. Then ϕ naturally induces a finite harmonic morphism of Λ-metrized complexes of curves

Σ(X ′, V ′ ∪D′) −→ Σ(X, V ∪D).

Remark 4.29. More generally, suppose that (X,V ∪D) and (X ′, V ′∪D′) are triangulated puncturedK-curves and that ϕ : X ′ → X is a finite morphism such that ϕ−1(D) = D′ and ϕ−1(V ) ⊂ V ′. Thenϕ−1(Σ) ⊂ Σ′ by Proposition 4.12(2). One can show using Theorem 4.23 that the map Σ′ → Xan

followed by the retraction Xan → Σ is a (not necessarily finite) harmonic morphism of metric graphsΣ′ → Σ.

4.30. Tame coverings of triangulated curves. Many of the results in this paper involve constructinga curveX ′ and a morphism ϕ : X ′ → X inducing a given morphism of skeleta as in Corollary 4.28. Forthese purposes it is useful to introduce some mild restrictions on the morphism ϕ. Fix a triangulatedpunctured curve (X,V ∪D) with skeleton Σ = Σ(X,V ∪D), regarded as a metrized complex of curvesas in (3.22).


Definition 4.31. Let (X ′, V ′∪D′) be a triangulated punctured curve with skeleton Σ′ = Σ(X ′, V ′∪D′)and let ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D) be a finite morphism. We say that ϕ is a tame covering of(X,V ∪D) provided that:

(1) D contains the branch locus of ϕ,(2) if char(k) = p > 0, then for every edge e′ ∈ E(Σ′) the expansion factor de′(ϕ) is not divisible

by p, and(3) ϕx′ is separable (= generically étale) for all x′ ∈ V ′.

Remark 4.32.

(1) Since D is finite, it follows from (1) and Theorem 4.23(3) that a tame covering of (X,V ∪D)is a tamely ramified morphism of curves.

(2) If Σ has at least one edge, then (2) implies (3).(3) Let ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D) be a tame covering, let S ⊂ Σ be a finite set of type-2

points, and let S′ = ϕ−1(S). Then S′ ⊂ Σ(X ′, V ′ ∪D′) is also a finite set of type-2 points, soϕ also defines a tame covering (X ′, S′ ∪ V ′ ∪D′)→ (X,S ∪ V ∪D).

Lemma 4.33. Let ϕ : (X ′, V ′ ∪D′) → (X,V ∪D) be a tame covering and let Σ = Σ(X,V ∪D) andΣ′ = Σ(X ′, V ′ ∪D′). Then ϕ|Σ′ : Σ′ → Σ is a tame covering of metrized complexes of curves.

Proof. Let Γ (resp. Γ′) be the augmented metric graph underlying Σ (resp. Σ′). By Proposition 2.22we only have to show that ϕ|Γ′ is generically étale. Let R =

∑x′∈V ′∪D′ Rx′ (x

′) be the ramificationdivisor of ϕ|Γ′ as defined in (2.14.1), and let S =

∑x′∈D′ Sx′ (x

′) be the ramification divisor ofϕ : X ′ → X. We will show that R = S. Since ϕx′ is generically étale for all x′ ∈ V ′, we seefrom (2.14.1) and the Riemann–Hurwitz formula as applied to ϕx′ that Rx′ ≥ 0. If x′ ∈ D′, then thereis a unique edge e′ adjacent to x′ and de′(ϕ) = dx′(ϕ), so

Rx′ = 2 dx′(ϕ)− 2− dx′(ϕ) + 1 = dx′(ϕ)− 1 = Sx′ ,

where the final equality is Theorem 4.23(3). Since Rx′ ≥ 0 for all x′ ∈ V ′, it is enough to show thatdeg(R) = deg(S). By (2.14.2) we have KΓ′ = (ϕ|Γ′)∗(KΓ) +R, so counting degrees gives

deg(R) = deg(ϕ|Γ′) (2− 2g(Γ))− (2− 2g(Γ′)).

The Lemma now follows from the equalities deg(ϕ) = deg(ϕ|Γ′), g(Γ) = g(X), g(Γ′) = g(X ′), and theRiemann–Hurwitz formula as applied to ϕ : X ′ → X. n

Remark 4.34. We showed in the proof of Lemma 4.33 that the ramification divisor of ϕ|Γ′ coincideswith the ramification divisor of ϕ. Moreover, it follows from Proposition 2.22 that for every x′ ∈ V ′,every ramification point x′ ∈ Cx′ is the reduction of a tangent direction in Γ′. In other words, for atame covering all ramification points of all residue curves are “visible” in the morphism of underlyingmetric graphs Γ′ → Γ.

Proposition 4.35. Let ϕ : (X ′, V ′ ∪D′)→ (X,V ∪D) be a finite morphism of triangulated puncturedcurves. Then ϕ is a tame covering if and only if

(1) D contains the branch locus of ϕ and(2) ϕx′ : Cx′ → Cϕ(x) is tamely ramified for every type-2 point x′ ∈ X ′an.

Moreover, if ϕ is a tame covering and x′ ∈ X ′an is a type-2 point not contained in Σ′, then ϕ is anisomorphism in a neighborhood of x′.

Proof. It is clear that conditions (1)–(2) imply that ϕ is a tame covering, so suppose that ϕ is atame covering and x′ ∈ X ′an is a type-2 point. We have the following cases:

• If x′ ∈ V ′, then ϕx′ is tamely ramified by Lemma 4.33 and Proposition 2.22.• Suppose that x′ is contained in the interior of an edge e′ of Σ′. Since ϕ is a tame covering of

(X, (V ∪ ϕ(y′)) ∪D) by Remark 4.32(3), we are reduced to the case treated above.• Suppose that x′ /∈ Σ′ but y′ = τ(x′) is contained in V ′. Let X (resp. X′) be the semistable

formal model of X (resp. X ′) corresponding to the semistable vertex set V (resp. V ′) and let


ϕ denote the unique finite morphism X′ → X extending ϕ : X ′ → X (see Theorem 5.13). Letx′ = red(x′) ∈ X′(k) and let x = ϕ(x′) and x = red(x) = ϕk(x′). The connected componentB′ of x′ in X ′an \ Σ′ is equal to red−1(x′), and the connected component B of x in Xan \Σ is equal to red−1(x). Since Σ ∩ B = ∅, ϕk is not branched over x by Lemma 4.33 andProposition 2.22. Therefore ϕ−1(B) is a disjoint union of deg(ϕ) open balls (one of which isB′) mapping isomorphically onto B. In particular, ϕ is an isomorphism in a neighborhood ofx′.

• Suppose that x′ /∈ Σ′ but y′ = τ(x′) is contained in the interior of an edge of Σ′. We claim thatϕ is an isomorphism in a neighborhood of x′. Let y = ϕ(y′) ∈ Σ. Then ϕ is a tame covering of(X, (V ∪y)∪D), so replacing V with V ∪y and V ′ with V ′∪ϕ−1(y) (see Remark 4.32(3)),we are reduced to the previous case.

n

Remark 4.36. It follows from Proposition 4.35 that if ϕ is a tame covering then the set-theoreticbranch locus of ϕ : X ′an → Xan (i.e. the set of all points in Xan with fewer than deg(ϕ) points in itsfiber) is contained in Σ. See [Fab13a, Fab13b] for more on the topic of the Berkovich ramificationlocus in the case of self-maps of P1.

5. APPLICATIONS TO MORPHISMS OF SEMISTABLE MODELS

In this section we show how Section 4 formally implies a large part of the results of Liu [Liu06]on simultaneous semistable reduction of morphisms of curves (see also [Col03]) over an arbitraryvalued field. In addition to establishing these results over a more general ground field, we feel thatthe “skeletal” point of view on simultaneous semistable reduction is enlightening (see Remark 5.23).To this end, we let K0 be any field equipped with a nontrivial non-Archimedean valuation val : K0 →R ∪ ∞. Its valuation ring will be denoted R0, its maximal ideal mR0

, and its residue field k0.Let X be a smooth, proper, geometrically connected algebraic K0-curve. By a (strongly) semistable

R0-model of X we mean a flat, integral, proper relative curve X → Spec(R0) whose special fiberXk0 is a (strongly) semistable curve (i.e. Xk0 is a reduced curve with at worst ordinary double pointsingularities; it is strongly semistable if its irreducible components are smooth) and whose genericfiber is equipped with an isomorphism XK0

∼= X. By properness of X , any K0-point x ∈ X(K0)extends in a unique way to an R0-point x ∈ X (R0); the special fiber of this point is the reduction ofx and is denoted red(x) ∈ X (k0). Let D ⊂ X(K0) be a finite set. A semistable model X of X is asemistable R0-model of (X,D) provided that the points of D reduce to distinct smooth points of Xk0 .The model X is a stable R0-model of (X,D) provided that every rational (resp. genus-1) componentof the normalization Xk contains at least three points (resp. one point) mapping to a singular point ofXk or to the reduction of a point of D.

If K0 = K is complete and algebraically closed, we define a (strongly) semistable formal R-model ofX to be an integral, proper, admissible formal R-curve X whose analytic generic fiber XK is equippedwith an isomorphism to Xan and whose special fiber Xk is a (strongly) semistable curve. There is anatural map of sets red : Xan → Xk, called the reduction; it is surjective and anti-continuous in thesense that the inverse image of a Zariski-open subset of Xk is a closed subset of Xan (and vice-versa).Using the reduction map, for a finite set D ⊂ X(K) we define semistable and stable formal R-modelsof (X,D) as above.

As we will be passing between formal and algebraic R-models of K-curves, it is worth stating thefollowing lemma; see [BPR11, Remark 5.30(2)].

Lemma 5.1. Let X be a smooth, proper, connected K-curve. The $-adic completion functor defines anequivalence from the category of semistable R-models of X to semistable formal R-models of X.

The inverse of the $-adic completion functor of Lemma 5.1 will be called algebraization.

5.2. Descent to a general ground field. We will use the following lemmas to descend the geometrictheory of Section 4 to the valued field K0. Fix an algebraic closure K0 of K0 and a valuation val on


K0 extending the given valuation on K0. For any field K1 ⊂ K0 we consider K1 as a valued field withrespect to the restriction of val, and we write R1 for the valuation ring of K1. Let K be the completionof K0 with respect to val. This field is algebraically closed by [BGR84, Proposition 3.4.1/3].

Lemma 5.3. Let X0 be a smooth, proper, geometrically connected curve over K0 and let X = X0⊗K0K.

Let Ksep0 be the separable closure of K0 in K. Then the image of X0(Ksep

0 ) under the natural inclusion

X0(Ksep0 ) ⊂ X0(K) = X(K) ⊂ Xan

is dense in Xan.Proof. Let K0 ⊂ K denote the completion of K0 and let (K0)sep ⊂ K be its separable closure.

By [BGR84, Proposition 3.4.1/6], (K0)sep is dense in K. Let K1 ⊂ (K0)sep be a finite, separableextension of K0. By Krasner’s lemma (see §3.4.2 of loc. cit.), there exists a finite, separable extensionK1/K0 contained in Ksep

0 which is dense in K1. It follows that Ksep0 is dense in (K0)sep, so Ksep

0 isdense in K. Since P1

K(K) is dense in P1,anK and the subspace topology on P1

K(K) ⊂ P1,anK coincides

with the ultrametric topology, this proves the lemma for X0 = P1K0

.For general X0, choose a finite, generically étale morphism ϕ : X0 → P1

K0. Let D ⊂ P1

K(K) bethe branch locus, so ϕ : Xan \ ϕ−1(D) → P1,an

K \D is étale, hence open by [Ber93, Corollary 3.7.4].It follows that if U ⊂ Xan is a nonempty open set, then ϕ(U) \ D contains a nonempty open subsetof P1,an

K \ D. Let x ∈ P1K0

(Ksep0 ) be a point contained in ϕ(U) \ D. Then ϕ−1(x) ⊂ X0(Ksep

0 ) andϕ−1(x) ∩ U 6= ∅. n

Lemma 5.4. Let X0 be a smooth, proper, geometrically connected curve over K0, let D ⊂ X0(K0) bea finite subset, let X = X0 ⊗K0 K, and let X be a semistable model of (X,D). There exists a finite,separable extension K1 of K0 and a semistable model X1 of X1 = X0⊗K0 K1 with respect to D such thatX1 ⊗R1

R ∼= X .Proof. Let X be the $-adic completion of X . For x ∈ Xk(k) the subset red−1(x) ⊂ Xan is open

in Xan. By Lemma 5.3, there exists a point x ∈ X0(Ksep0 ) with red(x) = x, so after passing to a

finite, separable extension of K0 if necessary, we may enlarge D ⊂ X0(K0) to assume that X is astable model of (X,D). On the other hand, by the stable reduction theorem (i.e. the properness ofthe Deligne-Mumford stack Mg,n parameterizing stable marked curves) there exists a stable modelX0 of (X0, D0) after potentially passing to a finite, separable extension of K0. By uniqueness of stablemodels we have X0 ⊗R0

R ∼= X . n

Lemma 5.5. Let X0, X′0 be smooth, proper, geometrically connected K0-curves and let ϕ : X ′0 → X0 be

a finite morphism. Let X0 (resp. X ′0) be a semistable model of X0 (resp. X ′0), let X = X0 ⊗K0 K andX = X0 ⊗R0 R (resp. X ′ = X ′0 ⊗K0 K and X ′ = X ′0 ⊗R0 R), and suppose that ϕK : X ′ → X extends toa morphism X ′ → X . Then ϕ extends to a morphism X ′0 → X0 in a unique way.

Proof. The morphism ϕ gives rise to a section X ′0 → X0 ×K0 X′0 of pr2 : X0 ×K0 X

′0 → X ′0. Let Z

be the schematic closure of the image of X ′0 in X0×R0X ′0. It is enough to show that pr2 : Z → X ′0 is an

isomorphism. Since schematic closure respects flat base change, the schematic closure of the imageof X ′ in X ×R X ′ is equal to Z ⊗R0

R. By properness of X ′, the image of X ′ in X ×R X ′ is then equalto Z ⊗R0 R, so pr2 : Z ⊗R0 R → X ′ is an isomorphism. Since R0 → R is faithfully flat, this impliesthat pr2 : Z → X ′0 is an isomorphism. n

5.6. The relation between semistable models and skeleta. Let X be a smooth, proper, connectedK-curve. The following theorem is due to Berkovich and Bosch-Lütkebohmert; see [BPR11, Theo-rem 5.34] for more precise references.

Theorem 5.7. Let X be a semistable formal model of X and let x ∈ Xk be a point. Then

(1) x is a generic point if and only if red−1(x) consists of a single type-2 point of Xan.(2) x is a smooth closed point if and only if red−1(x) is an open ball.(3) x is an ordinary double point if and only if red−1(x) is an open annulus.


It follows from Theorem 5.7 that if X is a semistable formal model of X, then

V (X) =

red−1(ζ) : ζ ∈ Xk is a generic point

is a semistable vertex set of X: indeed,

(5.7.1) X \ V (X) =∐x∈Xk

singular

red−1(x) ∪∐x∈Xksmooth

red−1(x)

is a disjoint union of open balls and finitely many open annuli. The following folklore theorem isproved in [BPR11, Theorem 5.38].

Theorem 5.8. Let X be a smooth, proper, connected algebraic K-curve and let D ⊂ X(K) be a finiteset of closed points. The association X 7→ V (X) sets up a bijection between the set of (strongly) semistableformal models of (X,D) (up to isomorphism) and the set of (strongly) semistable vertex sets of (X,D).

5.9. Let X be a semistable formal model of (X,D). One can construct the metrized complex ofcurves Σ(X,V (X) ∪ D) directly from X, as follows. Let V be the set of irreducible components ofXk, regarded as a set of vertices of a graph. For x ∈ V we will write Cx to denote the correspondingcomponent of Xk. To every double point x ∈ Xk we associate an edge e connecting the irreduciblecomponents of Xk containing x (this is a loop edge if there is only one such component); if Cx issuch a component, then we set redx(e) = x. The completed local ring of X at x is isomorphic toRJs, tK/(st − π) for some π ∈ mR \ 0; we define the length of e to be `(e) = val(π). We connecta point x ∈ D to the vertex corresponding to the irreducible component of Xk containing red(x).These data define a metrized complex of curves Σ(X, D). By (5.7.1), the graphs underlying Σ(X, D)and Σ(X,V (X) ∪D) are naturally isomorphic. The edge lengths coincide — see for instance [BPR11,Proposition 5.37] — and the residue curves of the two metrized complexes are naturally isomorphicby [Ber90, Proposition 2.4.4]. From this it is straightforward to verify that Σ(X, D) and Σ(X,V (X) ∪D) are identified as metrized complexes of curves.

In particular, a semistable formal model X of (X,D) is stable if and only if V (X) is a stable vertexset of (X,D).

As another consequence of Theorem 5.8, we obtain the following compatibility of skeleta andextension of scalars.

Proposition 5.10. Let X be a smooth, proper, connected K-curve, let K ′ be a complete and algebraicallyclosed valued field extension of K, and let π : Xan

K′ → XanK be the canonical map. If X is a semistable

formal model of X, then

(1) π maps V (XR′) bijectively onto V (X), and(2) π maps Σ(XK′ , V (XR′) ∪D) bijectively onto Σ(X,V (X) ∪D), with

π : Σ(XK′ , V (XR′) ∪D)∼−→ Σ(X,V (X) ∪D)

an isomorphism of augmented metric graphs.

Proof. The reduction map is compatible with extension of scalars, in that the following squarecommutes

XanK′

red //

π

Xk′

π

Xan

red// Xk

with k′ the residue field of K ′. The first assertion is immediate because the extension of scalarsmorphism π : Xk′ → Xk is a bijection on generic points. It is also a bijection on singular points, andfor any node x′ ∈ Xk′ the open annulus A′ = red−1(x′) is identified with the extension of scalars ofA = red−1(π(x′)). It is easy to see that π : A′ → A takes the skeleton of A′ isomorphically onto theskeleton of A. n


5.11. Extending morphisms to semistable models: analytic criteria. Let X be a smooth, proper,connected K-curve and let X be a semistable formal model of X. The inverse image topology on Xan

is the topology T (X) whose open sets are the sets of the form red−1(Uk), where Uk is a Zariski-opensubset of Xk. Note that any such set is closed in the natural topology on Xan. If Uk ⊂ Xk is an affineopen subset, then Uk is the special fiber of a formal affine open U = Spf(A) ⊂ X, the set underlyingthe generic fiber UK = M (AK) ⊂ Xan is equal to red−1(Uk), and A is equal to the ring AK of power-bounded elements in the affinoid algebraAK . This essentially means that X is a formal analytic varietyin the sense of [BL85]; see [BPR11, Remark 5.30(3)] for an explanation. Therefore X is constructedby gluing the canonical models of the affinoid subdomains of Xan which are open in the inverseimage topology T (X), along the canonical models of their intersections; here the canonical model ofan affinoid space M (B) is by definition Spf(B). The following general fact about formal analyticvarieties follows from these observations and the functoriality of the reduction map.

Proposition 5.12. Let X,X ′ be smooth, proper, connected algebraic curves over K, let ϕ : X ′ → X be afinite morphism, and let X and X′ be semistable formal models of X and X ′, respectively. Then ϕ extendsto a morphism X′ → X if and only if ϕ is continuous with respect to the inverse image topologies T (X)and T (X′), in which case there is exactly one morphism X′ → X extending ϕ.

Fix smooth, proper, connected algebraic K-curves X,X ′ and a finite morphism ϕ : X ′ → X. Thefollowing theorem is well-known to experts, although no proof appears in the literature to the best ofour knowledge.

Theorem 5.13. Let X and X′ be semistable formal models of X and X ′, respectively. Then ϕ : X ′ → Xextends to a morphism X′ → X if and only if ϕ−1(V (X)) ⊂ V (X′), and X′ → X is finite if and only ifϕ−1(V (X)) = V (X′).

Remark 5.14. If X and X′ are semistable formal models of the punctured curves (X,D) and (X ′, D′),respectively, and ϕ : X ′ → X is a finite morphism with ϕ−1(D) = D′ which extends to a morphismX′ → X, then since ϕ−1(V (X)) ⊂ V (X′) it follows from Remark 4.29 that there is a natural harmonicmorphism of metric graphs Σ(X ′, V (X′) ∪D′) → Σ(X,V (X) ∪D). The morphism X′ → X is finite ifand only if the local degree of this harmonic morphism at every v′ ∈ V (X′) is positive.

Before giving a proof of Theorem 5.13, we mention the following consequences. Let X1 and X2 besemistable formal models of X. We say that X1 dominates X2 provided that there exists a (necessarilyunique) morphism X1 → X2 inducing the identity map on analytic generic fibers. Taking X = X ′

in Theorem 5.13, we obtain the following corollary; this gives a different proof of the second partof [BPR11, Theorem 5.38].

Corollary 5.15. Let X1 and X2 be semistable formal models of X. Then X1 dominates X2 if and only ifV (X1) ⊃ V (X2).

In conjunction with Proposition 5.10, we obtain the following corollary. If K ′ is a complete andalgebraically closed field extension of K, we say that a semistable formal model X′ of XK′ is definedover R provided that it arises as the extension of scalars of a (necessarily unique) semistable formalmodel of X.

Corollary 5.16. Let K ′ be a complete and algebraically closed field extension of K, let X′ be a semistableformal model of XK′ , and suppose that there exists a semistable formal model X of X such that XR′

dominates X′. Then X′ is defined over R.Proof. Let π : Xan

K′ → Xan be the canonical map. By Corollary 5.15 we have V (XR′) ⊃ V (X′), andby Proposition 5.10 the map π defines a bijection V (XR′)

∼−→ V (X) and an isomorphism

Σ(XK′ , V (XR′))∼−→ Σ(X,V (X)).

Let V = V (X) and V ′ = π(V (X′)). We claim that V ′ is a semistable vertex set. Granted this,letting Y be the semistable formal model of X associated to V ′, we have V (YR′) = V (X′) sinceV (YR′) ⊂ V (XR′) (as XR′ dominates YR′) and π(V (YR′)) = V (Y), so YR′ = X′.


It remains to prove the claim that V ′ is a semistable vertex set. The point is that the question ofwhether or not a subset of V is a semistable vertex set is intrinsic to the augmented Λ-metric graphΣ = Σ(X,V ) ∼= Σ(XK′ , V (XR′)). We leave the details to the reader. n

We will use the following lemmas in the proof of Theorem 5.13. Recall from (3.11) that if V is asemistable vertex set of X, then a connected component C of Xan \ V is adjacent to a point x ∈ Vprovided that the closure of C in Xan contains x.

Lemma 5.17. Let X be a semistable formal model of X, let V = V (X), let Uk ⊂ Xk be a subset, and letU = red−1(Uk) ⊂ Xan. Then U is open in the topology T (X) if and only if the following conditions hold:

(1) U is closed in the ordinary topology on Xan, and(2) for every x ∈ U ∩ V , all but finitely many connected components of Xan \ V which are adjacent

to x are contained in U .

Proof. Let ζ be a generic point of Xk and let y ∈ Xk be a closed point. Let x ∈ V be the uniquepoint of Xan reducing to ζ and let B = red−1(y). By the anti-continuity of the reduction map, y is inthe closure of ζ if and only if x is adjacent to B. The lemma follows easily from this and the factthat the connected components of Xan \ V are exactly the inverse images of the closed points of Xkunder red. n

Lemma 5.18. Let V and V ′ be semistable vertex sets of X and X ′, respectively, and suppose thatϕ−1(V ) ⊂ V ′. Let C ′ be a connected component of X ′an \ ϕ−1(V ). Then C ′ has the following form:

(1) If C ′ intersects Σ(X ′, V ′), then C ′ = τ−1(Σ(X ′, V ′) ∩ C ′

), and

(2) otherwise C ′ is an open ball connected component of X ′an \ V ′.

Proof. Let Σ′ = Σ(X ′, V ′). Suppose that there exists y′ ∈ C ′ such that τ(y′) /∈ C ′. Let B′ be theconnected component of X ′an \ Σ′ containing y′, so B′ is an open ball contained in C ′ and τ(y′) isthe end of B′. It follows that τ(y′) is contained in the closure of C ′ in X ′an. Since C ′ is a connectedcomponent of X ′an \ ϕ−1(V ), its closure is contained in C ′ ∪ ϕ−1(V ), so τ(y′) ∈ ϕ−1(V ). ThereforeB′ = (B′ ∪ τ(y′)) ∩ (X ′an \ ϕ−1(V )) is open and closed in X ′an \ ϕ−1(V ), so B′ is a connectedcomponent of X ′an \ ϕ−1(V ) and hence B′ = C ′.

Now suppose that τ(C ′) ⊂ C ′. Then τ(C ′) ⊂ C ′ ∩ Σ′, so C ′ ⊂ τ−1(C ′ ∩ Σ′). Let y′ ∈ τ−1(C ′ ∩ Σ′)and let B′ be the connected component of X ′an \ Σ′ containing y′, as above. Since B′ = B′ ∪ τ(y′)is a connected subset of X ′an \ ϕ−1(V ) intersecting C ′ we have B′ ⊂ C ′, so y′ ∈ C ′ and thereforeC ′ = τ−1(C ′ ∩ Σ′). n

Proof of Theorem 5.13. Let V = V (X) and V ′ = V (X′). If there is an extension X′ → X of ϕ, thenthe square

(5.18.1) X ′an ϕ //

red

Xan

red

X′k

// Xk

commutes. Let x′ ∈ ϕ−1(V ), so x = ϕ(x′) reduces to a generic point ζ of Xk. Since the reduction ζ ′

of x′ maps to ζ, the point ζ ′ is generic, so x′ ∈ V ′. Therefore ϕ−1(V ) ⊂ V ′. The morphism X′ → X isfinite if and only if every generic point of X′k maps to a generic point of Xk; as above, this is equivalentto V ′ = ϕ−1(V ).

It remains to prove that if ϕ−1(V ) ⊂ V ′, then ϕ extends to a morphism X′ → X. By Proposi-tion 5.12, we must show that ϕ is continuous with respect to the topologies T (X) and T (X′). LetU ⊂ Xan be T (X)-open and let U ′ = ϕ−1(U). Clearly U is closed, so U ′ is closed (with respect to theordinary topologies). We must show that condition (2) of Lemma 5.17 holds for U ′. Let x′ ∈ U ′ ∩ V ′and let x = ϕ(x′). If x /∈ V , then let C ⊂ U be the connected component of Xan \ V containingx and let C ′ be the connected component of X ′an \ ϕ−1(V ) containing x′. Then C ′ ⊂ U ′ because


ϕ(C ′) ⊂ C, and C ′ contains every connected component of X ′an \ V ′ adjacent to x′ since C ′ is anopen neighborhood of x′.

Now suppose that x ∈ V . Any connected component of X ′an \ V ′ which is adjacent to x′ mapsinto a connected component of Xan \ V which is adjacent to x. There are finitely many connectedcomponents Xan \ V which are adjacent to x and not contained in U by Lemma 5.17. Let C besuch a component. Since ϕ is finite, there are only finitely many connected components of ϕ−1(C);each of these is a connected component of X ′an \ ϕ−1(V ). If C ′ is such a connected component,then either C ′ is an open ball connected component of X ′an \ V ′ or C ′ intersects Σ′ = Σ(X ′, V ′)by Lemma 5.18. There are finitely many connected components of X ′an \ V ′ which intersect Σ′ —these are just the open annulus connected components of X ′an \ V ′ — so there are only finitely manyconnected components of X ′an \ V ′ contained in ϕ−1(C). Therefore all but finitely many connectedcomponents of X ′an \V ′ which are adjacent to x′ map to connected components of Xan \V which arecontained in U . n

5.19. Simultaneous semistable reduction theorems. Recall that K0 is a field equipped with a non-trivial non-Archimedean valuation val : K0 → R ∪ ∞. As above we fix an algebraic closure K0 ofK0 and a valuation val on K0 extending the given valuation on K0; for any field K1 ⊂ K0 we considerK1 as a valued field with respect to the restriction of val, and we write R1 for the valuation ring of K1.In what follows, X and X ′ are smooth, proper, geometrically connected K0-curves and ϕ : X ′ → X isa finite morphism.

Proposition 5.20. Let X be a semistable R0-model of X. If there exists a semistable R0-model X ′ of X ′

such that ϕ : X ′ → X extends to a finite morphism X ′ → X , then there is exactly one such model X ′ upto (unique) isomorphism.

Proof. Suppose first that K = K0 is complete and algebraically closed. Let X (resp. X′) be the $-adic completion of X (resp. X ′). In this case, the proposition follows from Theorem 5.8 and the factthat if the morphism X′ → X of Theorem 5.13 is finite, then V (X′) is uniquely determined by V (X).The general case follows from this case after passing to the completion of the algebraic closure K ofK0: if X ′,X ′′ are two semistable models of X ′ such that ϕ extends to finite morphisms X ′ → X andX ′′ → X , then the isomorphism X ′′R

∼−→ X ′R descends to an isomorphism X ′′ ∼−→ X ′ by Lemma 5.5. n

Let X1,X2 be semistable R0-models of X. Recall that X1 dominates X2 if there exists a (necessarilyunique) morphism X1 → X2 inducing the identity on X.

Proposition 5.21. Let X and X ′ be semistable R0-models of X and X ′, respectively. Let D′ ⊂ X ′(K0)be a finite set of points. Then there exists a finite, separable extensionK1 ofK0 and a semistableR1-modelX ′′ of (X ′K1

, D′) such that:

(1) X ′′ dominates X ′R1,

(2) ϕK1 : X ′K1→ XK1 extends to a morphism X ′′ → XR1 , and

(3) any other semistable R1-model of (X ′K1, D′) satisfying the above two properties dominates X ′′.

Moreover, the formation of X ′′ commutes with arbitrary valued field extensions K1 → K ′1.Proof. Suppose first that K0 = K is complete and algebraically closed. Let X and X′ be the $-adic

completions of X and X ′, respectively. Let V = V (X) and V ′ = V (X′). By Lemma 3.15, there is aminimal semistable vertex set V ′′ of (X ′, D′) which contains ϕ−1(V ) ∪ V ′. Let X′′ be the semistableformal model of X ′ corresponding to V ′′. Then X′′ dominates X′ by Corollary 5.15 and ϕ extends to amorphism X′′ → X by Theorem 5.13. Part (3) follows from Corollary 5.15 and the minimality of V ′′.Taking X ′′ to be the algebraization of X′′ yields (1)–(3) in this case.

For a general valued field K0, suppose that K is the completion of K0. By Lemma 5.4 the model ofX ′K constructed above descends to a model X ′′ defined over the ring of integers of a finite, separableextension K1 of K0. Properties (1)–(3) follow from Lemma 5.5 and the corresponding properties ofX ′′R.

Now we address the behavior of this construction with respect to base change. Using Lemma 5.5we immediately reduce to the case of an extension K → K ′ of complete and algebraically closed


valued fields. Let R′ be the ring of integers of K ′. Let X ′′ (resp. X ′′′) be the minimal R-model of X ′

dominating X ′ (resp. R′-model of X ′K′ dominating X ′R′) mapping to X (resp. XR′). By (3) as appliedto X ′′′, we have that X ′′R′ dominates X ′′′. By Corollary 5.16, X ′′′ is defined over R, so by (3) as appliedto X ′′, we have X ′′R′ = X ′′′. n

Theorem 5.22. (Liu) Let X (resp. X ′) be a semistable R0-model of X (resp. X ′). Let D ⊂ X(K0) andD′ ⊂ X ′(K0) be finite sets, and suppose that ϕ(D′) ⊂ D. Then there exists a finite, separable extensionK1 of K0 and semistable R1-models X1,X ′1 of (XK1 , D), (X ′K1

, D′), respectively, such that

(1) X1 dominates XR1and X ′1 dominates X ′R1

,(2) the morphism ϕK1

: X ′K1→ XK1

extends to a finite morphism X ′1 → X1, and(3) if X2, X ′2 are semistable formal models of (XK1 , D), (X ′K1

, D′), respectively, satisfying (1)and (2) above, then X2 dominates X1 and X ′2 dominates X ′1.

Moreover, the formation of X ′1 → X1 commutes with arbitrary valued field extensions K1 → K ′1.

The morphism X ′1 → X1 is called the stable marked hull of X ′ 99K X in [Liu06].

Proof. First assume that K0 = K is complete and algebraically closed; let X,X′ be the $-adiccompletions of X ,X ′, respectively. Let V = V (X) and V ′ = V (X′). By Theorem 5.8, Theorem 5.13,and Corollary 5.15, we may equivalently formulate the existence and uniqueness of X′1 → X1 in termsof semistable vertex sets, as follows. We must prove that there exists a semistable vertex set V1 of(X,D) such that

(1) V1 contains V ∪ ϕ(V ′),(2) ϕ−1(V1) is a semistable vertex set of (X ′, D′), and(3) V1 is minimal in the sense that if V2 is another semistable vertex set of (X,D) satisfying (1)

and (2) above, then V2 ⊃ V1.

First we will prove the existence of V1 satisfying (1) and (2). By Lemma 3.15 we may enlarge Vto assume that V is a semistable vertex set of (X,D). Let Σ = Σ(X,V ∪ D) and Σ′ = Σ(X ′, V ′).By Corollary 4.18, there exists a skeleton Σ1 of (X,D) such that Σ1 ⊃ Σ ∪ ϕ(Σ′) and such thatΣ′1 = ϕ−1(Σ1) is a skeleton of (X ′, ϕ−1(D)). Let V ′1 be a vertex set for Σ′1. Since any finite subset oftype-2 points of Σ′1 which contains V ′1 is again a vertex set for Σ′1 by Proposition 3.12(4), we may anddo assume that V ′ ⊂ V ′1 . Let V1 ⊂ Σ1 be the union of a vertex set for Σ1 with V ∪ ϕ(V ′1). Then V1 isa semistable vertex set of (X,D), and ϕ−1(V1) ⊂ Σ′1 is a finite set of type-2 points containing V ′1 , thusis a semistable vertex set of (X ′, ϕ−1(D)) (hence of (X ′, D′) as well).

To prove that there exists a minimal such V1, we make the following recursive construction. LetV (0) = V , let V ′(0) = V ′, and for each n ≥ 1 let V (n) be the minimal semistable vertex set of (X,D)containing V (n − 1) ∪ ϕ(V ′(n − 1)) and let V ′(n) be the minimal semistable vertex set of (X ′, D′)containing ϕ−1(V (n)). These sets exist by Lemma 3.15. By induction it is clear that if V2 is anysemistable vertex set of (X,D) satisfying (1) and (2) above, then V (n) ⊂ V2 for each n. Since V2

is a finite set, for some n we have V (n) = V (n + 1), which is to say that ϕ(V ′(n)) ⊂ V (n); sinceV ′(n) ⊃ ϕ−1(V (n)), we have that V ′(n) = ϕ−1(V (n)) is a semistable vertex set of (X ′, D′). ThenV1 = V (n) is the minimal semistable vertex set satisfying (1) and (2) above.

The case of a general ground field reduces to the geometric case handled above exactly as in theproof of Proposition 5.21, as does the statement about the behavior of the stable marked hull withrespect to valued field extensions. n

Remark 5.23. (The skeletal viewpoint on Liu’s theorem) The statement of Theorem 5.22 is stronglyanalogous to Corollary 4.18, which is indeed the main ingredient in the proof. The difference is thatwhereas finite morphisms of semistable models correspond to pairs V, V ′ of semistable vertex sets suchthat ϕ−1(V ) = V ′, for finite morphisms of triangulated punctured curves one requires in addition thatϕ−1(Σ) = Σ′. The former condition (of Liu’s theorem) does not imply the latter: for instance, letX ′ be the Tate curve y2 = x3 − x + $, let X = P1, and let ϕ : X ′ → X be the cover (x, y) 7→ x.This extends to a finite morphism of semistable models given by the same equations; however, theassociated skeleton of X ′ (resp. of P1) is a circle (resp. a point), so ϕ−1(Σ) ( Σ′.


In general, if ϕ−1(V ) = V ′ and ϕ−1(D) = D′, then by Remark 4.19 the image of the skeletonΣ(X ′, V ′ ∪D′) is equal to Σ(X,V ∪D) union a finite number of geodesic segments T1, . . . , Tr, and byTheorem 4.13 the saturation ϕ−1(ϕ(Σ(X ′, V ′ ∪D′)) is a skeleton of X ′.

Theorem 5.25 below follows the same philosophy in deriving a simultaneous semistable reductiontheorem of Liu–Lorenzini from Proposition 4.12.

Remark 5.24. Liu in fact works over an arbitrary Dedekind scheme (a connected Noetherian regularscheme of dimension 1), which includes discrete valuation rings but not more general valuation rings.In his statement of Theorem 5.22 the given models X ,X ′ are allowed to be any integral, projectiveSpec(R0)-schemes with generic fibers X and X ′, respectively. Although we restrict to semistable mod-els X ,X ′ in this paper, using a more general notion of a triangulation our methods can be extendedto treat relatively normal models X ,X ′.

The following simultaneous stable reduction theorem can be found in [LL99]. Theorem 5.25 is toProposition 4.12 as Theorem 5.22 is to Corollary 4.18

Theorem 5.25. (Liu–Lorenzini) Suppose that (X,D) and (X ′, D′) are both stable and that ϕ−1(D) =D′. Assume that (X,D) and (X ′, D′) admit stable models X and X ′, respectively, defined over R0. Thenϕ : X ′ → X extends to a (not necessarily finite) morphism X ′ → X .

Proof. By Lemma 5.5 we may assume that K0 = K is complete and algebraically closed. Let V(resp. V ′) be the minimal semistable vertex set of (X,D) (resp. (X ′, D′)). By Theorem 5.13, we mustshow that ϕ−1(V ) ⊂ V ′. Let Σ = Σ(X,V ) and Σ′ = Σ(X ′, V ′). Let V1 be a vertex set for Σ withrespect to which Σ has no loop edges. Recall that

V = x ∈ V1 : g(x) ≥ 1 or x has valency at least 3

by Proposition 3.18(2). Let V ′1 = ϕ−1(V1) ∪ V ′; this is a vertex set for Σ′ since ϕ−1(Σ) ⊂ Σ′ byProposition 4.12(2). If x ∈ V has genus at least 1, then any x′ ∈ ϕ−1(x) has genus at least 1, soϕ−1(x) ⊂ V ′. Let x ∈ V have genus 0, so x has valency at least 3 in Σ. Let x′ ∈ ϕ−1(x) ⊂ Σ′ and letU ′ be the set containing x′ and all of the connected components of X ′an \ V ′1 adjacent to x′. Then U ′

is an open neighborhood of x′, so ϕ(U ′) is an open neighborhood of x by [Ber90, Lemma 3.2.4]. LetA be an open annulus connected component of Xan \ V1 adjacent to x. Any open neighborhood of xintersects A, so there exists y′ ∈ U ′ with ϕ(y′) ∈ A. Let A′ be the connected component of U ′ \ x′containing y′, so ϕ(A′) ⊂ A. Since x′ is an end of A′ and ϕ(x′) is an end of A, Lemma 4.3 implies thatA′ is an open annulus. Since x has valency at least 3 in Σ (and since Σ has no loop edges), there areat least 3 distinct open annulus connected components of Xan \ V1 adjacent to x. By the above, thesame is true of x′, so x′ has valency at least 3 in Σ′, so x′ ∈ V ′, as desired. n

6. A LOCAL LIFTING THEOREM

We now begin with the classification of lifts of harmonic morphisms of metrized complexes ofcurves to finite morphisms of algebraic curves. We begin by working in the neighborhood of a vertexof a metrized complex.

6.1. Let X be a smooth, proper, connected curve over K and let x ∈ Xan be a type-2 point. LetV ⊂ Xan be a strongly semistable vertex set of X containing x, let Σ = Σ(X,V ), and let τ : Xan → Σbe the retraction. Let e1, . . . , er be the edges of Σ adjacent to x and let Σ0 = x ∪ e1 ∪ · · · ∪ er ,where for a (closed) edge e we let e denote the corresponding open edge, i.e. the edge without itsendpoints. Then Σ0 is an open neighborhood of x in Σ and τ−1(Σ0) is an open neighborhood of xin Xan. Following [BPR11, 5.54] and [Ber93], we define a simple neighborhood of x to be an openneighborhood of this form (for some choice of V ). The connected components of τ−1(Σ0) \ x areopen balls and the open annuli τ−1(e1), . . . , τ−1(er).

Definition 6.2. A star-shaped curve is a pointed K-analytic space (Y, y) which is isomorphic to (U, x)where x is a type-2 point in the analytification of a smooth, proper, connected curve over K and U isa simple neighborhood of x. The point y is called the central vertex of Y .


6.3. Let (Y, y) be a star-shaped curve, so Y \ y is a disjoint union of open balls and finitely manyopen annuli A1, . . . , Ar. The skeleton of Y is defined to be the set Σ(Y, y) = y ∪

⋃ri=1 Σ(Ai).

A compatible divisor in Y is a finite set D ⊂ Y (K) whose points are contained in distinct open ballconnected components of Y \ y, so the connected components of Y \ (y ∪D) are open balls, theopen annuli A1, . . . , Ar, and (finitely many) open balls B1, . . . , Bs punctured at a point of D. Thedata (Y, y,D) of a star-shaped curve along with a compatible divisor is called a punctured star-shapedcurve. The skeleton of (Y, y,D) is the set

Σ(Y, y ∪D) = y ∪D ∪r⋃i=1

Σ(Ai) ∪s⋃j=1

Σ(Bj).

Fix a compatible divisorD in Y , and let Σ0 = Σ(Y, y∪D). There is a canonical continuous retractionmap τ : Y → Σ0 defined exactly as for skeleta of algebraic curves (3.11). The connected componentsof Σ0 \ y are called the edges of Σ0; an edge is called infinite if it contains a point of D and finiteotherwise.

If Y is the simple neighborhood τ−1(Σ0) of x ∈ Xan as above, then Σ(Y, x) = Σ0 = Σ ∩ Y . Afinite set D ⊂ Y (K) is compatible with Y if and only if V is a semistable vertex set for (X,D), inwhich case Σ(Y, x ∪D) = Y ∩Σ(X,V ∪D). The retraction τ : Y → Σ(Y, x ∪D) is the restrictionof the canonical retraction τ : Xan → Σ(X,V ∪D).

6.4. Let (Y, y) be a star-shaped curve. Then Y is proper as a K-analytic space if and only if allconnected components of Y \ y are open balls, i.e. if and only if Σ(Y, y) = y. If Y is proper,then there is a smooth, proper, connected curve X over K and an isomorphism f : Y

∼−→ Xan. Letx = f(y). Then x is a semistable vertex set of X, so by Theorem 5.8 there is a unique smooth formalmodel X of Xan such that x reduces to the generic point of Xk. Let D ⊂ Y (K) be a finite set. Then Dis compatible with Y if and only if the points of f(D) reduce to distinct closed points of Xk.

Conversely, let X be a smooth, proper, connected formal curve over Spf(R). If x ∈ Xan is the pointreducing to the generic point of Xk, then (XK , x) is a proper star-shaped curve.

Proposition 6.5. Let (Y, y) be a star-shaped curve. Then (Y, y) is isomorphic to (U, x) where x is thecentral vertex of a proper star-shaped curve X and U is a simple neighborhood of x.

Proof. LetA1, . . . , Ar be the open annulus connected components of Y \y. Choose isomorphismsfi : Ai

∼−→ S(ai)+ with standard open annuli such that fi(x) approaches the Gauss point of B(1) as xapproaches y. Let X be the curve obtained from Y by gluing an open ball B(1)+ onto each Ai via theinclusions fi : Ai

∼−→ S(ai)+ ⊂ B(1)+. Then X is a smooth, proper, connected K-analytic curve, andit is clear from the construction that (X, y) is star-shaped and that Y is a simple neighborhood of y inX. n

6.6. A proper star-shaped curve X and an inclusion i : Y∼−→ U ⊂ X as in Proposition 6.5 is called a

compactification of Y (as a star-shaped curve). Note that X \ i(Y ) is a disjoint union of finitely manyclosed balls, one for each open annulus connected component of Y \ y.6.7. Let (Y, y) be a star-shaped curve. The smooth, proper, connected k-curve Cy with function fieldH (y) is called the residue curve of Y . The tangent vectors in Ty are naturally in bijective correspon-dence with the connected components of Y \y. We define a reduction map red : Y → Cy by sendingy to the generic point of Cy, and sending every point in a connected component B of Y \ y to theclosed point of Cy corresponding to the tangent vector determined by B. This sets up a one-to-onecorrespondence between the connected components of Y \ y and the closed points of Cy.

Remark 6.8.

(1) When Y is proper, so Y ∼= XK for a smooth, proper, connected formal curve over Spf(R),then Cy ∼= Xk and the reduction map Y → Cy coincides with the canonical reduction mapXK → Xk.


(2) Let D ⊂ Y (K) be a compatible divisor and let Σ0 = Σ(Y, y ∪ D). Every edge e of Σ0 iscontained in a unique connected component of Y \y, and red(e) ∈ Cy(k) is the closed pointcorresponding to the tangent direction represented by e.

6.9. Tame coverings. We now study a class of morphisms of star-shaped curves analogous to (4.30).We begin with the following technical result.

Lemma 6.10. Let A,A′ be open annuli or punctured open balls and let ϕ : A′ → A be a finite morphismof degree δ. Suppose that δ is prime to char(k) if char(k) > 0. Fix an isomorphism A ∼= S(a)+ (where weallow a = 0).

(1) There is an isomorphism A′ ∼= S(a′)+ such that the composition

(6.10.1) S(a′)+∼= A′

ϕ−→ A ∼= S(a)+

is t 7→ tδ.(2) There is an isomorphism A′ ∼= S(a′)+ such that (6.10.1) extends to a morphism ψ : B(1)+ →

B(1)+ with ψ−1(0) = 0.(3) If (6.10.1) extends to a morphism ψ : B(1)+ → B(1)+ for a given isomorphism A′ ∼= S(a′)+,

then the extension is unique.

Proof. Let u be a parameter on A′, i.e. an isomorphism u : A′∼−→ S(a′)+ ⊂ B(1)+ with a standard

open annulus. By Lemma 4.2, ϕ restricts to an affine map on skeleta Σ(A′) → Σ(A) with degreeδ. If B = S(b, c) is a closed sub-annulus of S(a′)+, then by [Thu05, Lemme 2.2.1], after potentiallyreplacing u by u−1, ϕ∗(t) has the form αuδ(1 + g(u)) on B, where α ∈ R× and |g|sup < 1. Since δis not divisible by char(k) if char(k) > 0, the Taylor expansion for δ

√1 + g has coefficients contained

in R, hence converges to a δth root of 1 + g(u) on A. Choosing a δth root of α as well and lettinguB = u δ

√α δ√

1 + g, we have that uB is a parameter on B such that ϕ∗(t) = uδB . Hence for each suchB there are exactly δ choices of a parameter uB on B such that ϕ∗(t) = uδB; choosing a compatibleset of such parameters for all B yields a parameter on A′ satisfying (1).

Part (2) follows immediately from (1). A morphism from a K-analytic space X to B(1)+ is givenby a unique analytic function f on X such that |f(x)| < 1 for all x ∈ X, so (3) follows from the factthat the restriction homomorphism O(B(1)+)→ O(S(a′)+) is injective. n

6.11. Let A be an open annulus. Let δ be a positive integer, and assume that δ is prime to thecharacteristic of k if char(k) > 0. Let A′ be an open annulus and let ϕ : A′ → A be a finite morphismof degree δ. By Lemma 6.10, the group AutA(A′) is isomorphic to Z/δZ. Moreover, if A′′ is anotheropen annulus and ϕ′ : A′′ → A is a finite morphism of degree δ, then there exists an A-isomorphismψ : A′

∼−→ A′′; the induced isomorphism AutA(A′)∼−→ AutA(A′′) is independent of the choice of

ψ since both groups are abelian. Therefore the group AutA(δ) B AutA(A′) ∼= Z/δZ is canonicallydetermined by A and δ.

Definition 6.12. Let (Y, y,D) be a punctured star-shaped curve with skeleton Σ0 = Σ(Y, y ∪ D)and let Cy be the residue curve of Y . A tame covering of (Y, y,D) consists of a punctured star-shapedcurve (Y ′, y′, D′) and a finite morphism ϕ : Y ′ → Y satisfying the following properties:

(1) ϕ−1(y) = y′,(2) D′ = ϕ−1(D), and(3) if Cy′ denotes the residue curve of Y ′ and ϕy′ : Cy′ → Cy is the morphism induced by ϕ, then

ϕy′ is tamely ramified and is branched only over the points of Cy corresponding to tangentdirections at y represented by edges in Σ0.

Remark 6.13. The degree of ϕ is equal to the degree of ϕy′ .

Example 6.14. Let ϕ : (X ′, V ′, D′)→ (X,V,D) be a tame covering of triangulated punctured curves.Let Σ = Σ(X,V ∪D) and Σ′ = Σ(X ′, V ′∪D′) = ϕ−1(Σ). Let x ∈ V be a finite vertex of Σ, let e1, . . . , erbe the finite edges of Σ adjacent to x, let Σ0 = x ∪

⋃ri=1 e

i , let Y = τ−1(Σ0), and let D0 = D ∩ Y .

Then (Y, x,D0) is a punctured star-shaped curve. Let Σ′0 be a connected component of ϕ−1(Σ0), let


x′ ∈ Σ′0 be the unique inverse image of x, let Y ′ = τ−1(Σ′0), and let D′0 = D′ ∩ Y ′. Then (Y ′, x′, D′0)is also a punctured star-shaped curve, and ϕ restricts to a finite morphism ϕ : Y ′ → Y . This is in facta tame covering of punctured star-shaped curves by Remark 4.34 and Proposition 4.35.

Proposition 6.15. Let ϕ : (Y ′, y′, D′)→ (Y, y,D) be a degree-δ tame covering of punctured star-shapedcurves.1 Let Σ0 = Σ(Y, y ∪D), let Σ′0 = Σ(Y ′, y′ ∪D′), let Cy (resp. Cy′) be the residue curve of Y(resp. Y ′), and let ϕy′ : Cy′ → Cy be the induced morphism.

(1) Let B be a connected component of Y \ y disjoint from Σ0. Then ϕ−1 is a disjoint union of δopen balls mapping isomorphically onto B.

(2) Let B be a connected component of Y \ y meeting D, and choose an isomorphism of B withB(1)+ which identifies the unique point of B ∩ D with 0. Let B′ be a connected componentof ϕ−1(B). Then ϕ|B′ : B′ → B is a finite morphism, the degree of ϕ|B′ is the ramificationdegree δ′ of ϕy′ at red(B′), and there is an isomorphism B′ ∼= B(1)+ sending the unique pointof B′ ∩D′ to 0 such that the composition

B(1)+∼= B′

ϕ−→ B ∼= B(1)+

is t 7→ tδ′.

(3) Let A be an open annulus connected component of Y \ y, and choose an isomorphism A ∼=S(a)+. Let A′ be a connected component of ϕ−1(A). Then ϕ|A′ : A′ → A is a finite morphism,the degree of ϕ|A′ is the ramification degree δ′ of ϕy′ at red(A′), and there is an isomorphism ofA′ with an open annulus S(a′) such that the composition

S(a′)+∼= A′

ϕ−→ A ∼= S(a)+

is t 7→ tδ′.

(4) ϕ is étale over Y \D.(5) ϕ−1(Σ0) = Σ′0.

Proof. In the situation of (1), let x = red(B). Since ϕy′ is not branched over x, there are δ distinctpoints of Cy′ mapping to x; hence there are δ connected components B′1, . . . , B

′δ of Y ′ \ y′ mapping

onto B. The restriction of ϕ to each B′i is finite of degree 1.Next we prove (2). It is clear that ϕ|B′ : B′ → B is finite, hence surjective; since B′ is a connected

component of Y ′ \ y′ containing a point of D′ = ϕ−1(D), it is an open ball. Let x (resp. x′) be theunique point of D (resp. D′) contained in B (resp. B′) and let e ⊂ Σ0 (resp. e′ ⊂ Σ′0) be the edgeadjacent to x (resp. x′). Then B′ \ x′ → B \ x is a finite morphism of punctured open balls, so byLemma 3.3, Proposition 4.8, and Theorem 4.23(2), the restriction of ϕ to e′ is affine morphism e′ → eof degree δ′, and δ′ is the degree of B′ → B. So by Lemma 6.10 there is an isomorphism B′ ∼= B(1)+

as described in the statement of the Theorem.In the situation of (3), we claim that A′ is an open annulus. Clearly ϕ|A′ : A′ → A is finite,

hence surjective. If A′ is not an open annulus, then A′ ∼= B(1)+ is an open ball. The morphismB(1)+

∼= A′ → A ∼= S(a)+ is given by a unit on B(1)+, which has constant absolute value; thiscontradicts surjectivity, so A′ is in fact an open annulus. The proof now proceeds exactly as above.

Parts (4) and (5) follow immediately from parts (1)–(3). n

The following proposition is reminiscent of [Saï04, Proposition 3.3.2].

Corollary 6.16. Let ϕ : (Y ′, y′, D′0) → (Y, y,D0) be a tame covering of punctured star-shaped curvesand let i : (Y, y) → (X,x) be a compactification of Y . Let D1 be the union of i(D0) with a choice ofK-point from every connected component of X \ i(Y ). Then there exists a compactification i′ : (Y ′, y′) →(X ′, x′) and a tame covering ψ : (X ′, x′, D′1)→ (X,x,D1) such that ψ i′ = i ϕ.

Proof. We compactify (Y ′, y′) as in the proof of Proposition 6.5, gluing balls onto the annulusconnected components of Y ′\y′. By Proposition 6.15(3), ifA ∼= S(a)+ is an open annulus connectedcomponent of Y \y and A′ ⊂ Y ′ \y′ is a connected component mapping to A, then we can choose

1This δ need not be prime to char(k).


an isomorphism A′ ∼= S(a′)+ such that S(a′)+∼= A′ → A ∼= S(a)+ is of the form t 7→ tδ; this map

extends to a morphism B(1)+ → B(1)+, and these maps glue to give a tame covering X ′ → X. n

6.17. A local lifting theorem. Let (Y, y,D) be a punctured star-shaped curve with skeleton Σ0 =Σ(Y, y∪D) and residue curve Cy. Let C ′ be a smooth, proper, connected k-curve and let ϕ : C ′ → Cybe a finite, tamely ramified morphism branched only over the points of Cy corresponding to tangentdirections at y represented by edges in Σ0. A lifting of C ′ to a punctured star-shaped curve over (Y, y,D)is the data of a punctured star-shaped curve (Y ′, y′, D′), a tame covering ϕ : (Y ′, y′, D′) → (Y, y,D),and an isomorphism of the residue curve Cy′ with C ′ which identifies ϕwith the morphism ϕy′ : Cy′ →Cy induced by ϕ. An isomorphism between two liftings (Y ′, y′) and (Y ′′, y′′) is a Y -isomorphismY ′ → Y ′′ such that the induced morphism Cy′

∼−→ Cy′′ respects the identifications Cy′ ∼= C ′ andCy′′ ∼= C ′.

Theorem 6.18. Let (Y, y,D) be a punctured star-shaped curve with skeleton Σ0 = Σ(Y, y ∪ D)and residue curve Cy, let C ′ be a smooth, proper, connected k-curve, and let ϕ : C ′ → Cy be a finite,tamely ramified morphism branched only over the points of Cy corresponding to tangent directions aty represented by edges in Σ0. Then there exists a lifting of C ′ to a punctured star-shaped curve over(Y, y,D), and this lifting is unique up to unique isomorphism.

Before giving the proof of this theorem, we need a technical lemma. Let X be a finitely pre-sented, flat, separated R-scheme and let X be its $-adic completion; this is an admissible formalR-scheme by [BPR11, Proposition 3.12]. There is a canonical open immersion iX : XK → X an

K de-fined in [Con99, §A.3] which is functorial in X and respects the formation of fiber products, andis an isomorphism when X is proper over R. (This fact is implicitly contained in the statement ofLemma 5.1.)

Lemma 6.19. Let X ,X ′ be finitely presented, flat, separated R-schemes and let X,X′ denote their$-adic completions. Let X ′ → X be a finite and flat morphism. Then the square

(6.19.1) X′KiX′ //

f

X ′anK

g

XK

iX// X anK

is Cartesian.Proof. The vertical arrows f, g of (6.19.1) are finite and the horizontal arrows iX ′ , iX are open

immersions. Let Y = X ′anK ×X an

KXK , so Y → X ′an

K is an open immersion and Y → XK is finite.Let h : X′K → Y be the canonical morphism. Then h is an open immersion because its compositionwith Y → X ′an

K is an open immersion, and h is finite because its composition with Y → XK is finite.Therefore h is an isomorphism of X′K onto an open and closed subspace of Y . It suffices to show thath is surjective, and since Y (K) is dense in Y , we only need to check that Y (K) is in the image of h.

Let x ∈ XK(K) and y ∈ X ′anK (K) be points with the same image z ∈ X an

K (K), so (x, y) ∈ Y (K).Choose an open formal affine U ⊂ X such that x ∈ UK(K), and assume without loss of generality thatthere is an affine open U = Spec(A) ⊂ X such that U = Spf(A), where A is the $-adic completion ofA. Then x corresponds to a continuous K-homomorphism η : A ⊗R K → K. We have η(A) ⊂ R, socomposing with the completion homomorphism A → A we obtain an R-homomorphism A → R. Letξ : Spec(R) → U ⊂ X denote the induced morphism. The corresponding morphism Spf(R)K → XKis the point x.

Let Z be the fiber product of X ′ → X with η : Spec(R) → X and let Z be its $-adic completion.Note that Z is finite and flat over Spec(R). In particular, Z is proper over R, so iZ : ZK → Zan

K is an


isomorphism. We have a commutative cube

ZanK

//

X ′anK

g

ZK

∼=66

//

X′K

<<

f

Spec(K)an z // X anK

Spf(R)K

∼= 66

x // XK

<<

where the front and back faces are Cartesian and the diagonal arrows are the canonical open immer-sions. Since g(y) = z and Zan

K = f−1(z), the point y lifts to a point in ZanK (K), which then lifts to a

point in ZK(K) whose image in X′K(K) maps to x in XK(K) and y ∈ X ′anK (K). n

6.20. Proof of Theorem 6.18. We first prove the theorem when Y is proper. In this case we mayand do assume that Y is the analytic generic fiber of a smooth, proper, connected formal curve X overSpf(R). Let X → Spec(R) be the algebraization of X (Lemma 5.1); this is a smooth, proper relativecurve of finite presentation and with connected fibers whose $-adic completion is isomorphic to X.Note that Xk = Xk = Cy. Let X = XK , so Xan = XK = Y . By the valuative criterion of properness,every point x of D extends uniquely to a section Spec(R) → X which sends the closed point to thereduction of x; hence the closure D of D in X is a disjoint union of sections. Let U = X \ D.

The theory of the tamely ramified étale fundamental group πt1 of a morphism of schemes with a rel-ative normal crossings divisor is developed in [SGA1, Exposé XIII]. The finite-index subgroups of πt1classify so-called tamely ramified étale covers. The subscheme D ⊂ X is a relative normal crossingsdivisor relative to Spec(R), so the proof of Corollaire 2.12 of loc. cit. shows that the specializationhomomorphism π1(UK) → πt1(Uk) is surjective (we suppress the base points). What this means con-cretely is that every tamely ramified cover U ′ → Uk over Xk relative to Dk extends to a finite étalemorphism U ′ → U , unique up to unique isomorphism, and the generic fiber of U ′ is connected.

Let D′ = ϕ−1(Dk) and let U ′ = C ′ \ D′, so U ′ → Uk is a tamely ramified cover of Uk over Xkrelative to Dk. Let ϕ : U ′ → U be the unique étale covering extending ϕ|U ′ ; this is equipped with anisomorphism U ′k ∼= U ′ identifying ϕ|U ′ with ϕk. Let U = UK = X\D, let U ′ = U ′K , letX ′ be the smoothcompactification of U ′, and let ϕK : X ′ → X denote the unique morphism extending ϕK : U ′ → U .We will show that there is a simple neighborhood W of y in Xan such that ϕ−1

K (W ) → W is a tamecovering, and conclude that a lifting exists using Corollary 6.16.

6.20.1. First we claim that ϕ−1K (y) consists of a unique point y′ ∈ X ′an. Letting U and U′ denote the

$-adic completions of U and U ′, respectively, we have a Cartesian square

U′KiU′ //

U ′an

UK

iU// Uan

by Lemma 6.19. In other words, U′K is the inverse image of UK under ϕK : U ′an → Uan. Since thereduction map UK → Uk = Uk is surjective and y is the unique point of Xan reducing to the genericpoint of Uk ⊂ Xk, we have y ∈ UK , so it suffices to show that there is a unique point y′ ∈ U′K mappingto y. It follows easily from the functoriality of the reduction map that the only point y′ mapping to yis the unique point of U′K reducing to the generic point of U′k. In particular, the residue curve of X ′an

at y′ is identified with C ′, which is the smooth completion of U′k = U ′.

6.20.2. Choose a strongly semistable vertex set V ′ of X ′ containing y′, let Σ′ = Σ(X ′, V ′), and letτ : X ′an → Σ′ be the retraction. Let B ⊂ Xan be the formal fiber of a point x ∈ Uk(k); equivalently,B is a connected component of Xan \ y not meeting D. Let B′ be a connected component of


X ′an \ y′ contained in ϕ−1K (B). We have B ⊂ UK ; therefore B′ ⊂ U′K , so B′ is a connected

component of U′K \ y′, hence B′ is the formal fiber of a point x′ ∈ U′k(k), so B′ ∼= B(1)+. It followsthat V ′ \ (V ′ ∩B′) is again a semistable vertex set, so we may and do assume that Σ′ ∩B′ = ∅ for allsuch B′.

Let e′ ⊂ Σ′ be an edge adjacent to y′ and let B be the connected component of Xan\y containingϕK(e′). By the above, B is the formal fiber of a point x ∈ Dk. Fix an isomorphism B ∼= B(1)+ takingthe unique point in D ∩ B to 0. Let v′ ∈ Ty′ be the tangent vector in the direction of e′ and letv = dϕK(y′)(v′) ∈ Ty. The point in Xk(k) corresponding to v is x; let x′ ∈ D′ ⊂ C ′(k) be the pointcorresponding to v′, so ϕ(x′) = x. By Theorem 4.23, the ramification degree δ of ϕ at x′ is equal todv′ϕK(x′).

Let B′ be the connected component of X ′an \ y′ containing e′. Viewing ϕK |B′ as a morphismB′ → B(1)+ ⊂ A1,an

K via our chosen isomorphism B ∼= B(1)+, the map x′ 7→ log |ϕK(x′)| is apiecewise affine function on e′ which changes slope on the retractions of the zeros of ϕK . Shrinkinge′ if necessary, we may and do assume that log |ϕK | is affine on e′; its slope has absolute value δ, andlog |ϕK(x′)| → 0 as x′ → y′. By (4.24) the restriction of ϕK to the open annulus τ−1(e′) is a finitemorphism of degree δ onto an open annulus S(a)+ ⊂ B(1)+, where val(a) is δ times the length of e′.

6.20.3. Enlarging Σ′ if necessary, we may and do assume that every tangent direction v′ ∈ Ty′ corre-sponding to a point in D′ is represented by an edge in Σ′. Let e′1, . . . , e

′r be the edges of Σ′ adjacent

to y′, for i = 1, . . . , r let Bi be the connected component of Xan \ y containing ϕK(e′i ), and chooseisomorphisms Bi

∼−→ B(1)+ sending the unique point of D ∩ Bi to 0. (The balls B1, . . . , Br are notnecessarily distinct; we mean that one should choose a single isomorphism for each distinct ball.) Letx′i ∈ D′ be the point corresponding to the tangent direction at y′ in the direction of e′i, and let δi bethe ramification degree of ϕ at x′i. Applying the procedure of (6.20.2) for each e′i, and shrinking ifnecessary, we may and do assume that for every i, ϕK induces a degree-δi morphism of τ−1(e′i ) ontoan open annulus S(a)+ ⊂ B(1)+

∼= Bi, with a independent of i.Let W ⊂ Xan be the union of τ−1(y) with the annuli S(a)+ ⊂ Bi, so W is obtained from Xan by

removing a closed ball around each point of D. This is a simple neighborhood of y in Xan, hence isa star-shaped curve. Let Σ′0 = y′ ∪

⋃ri=1 e

′i and let W ′ = τ−1(Σ′0), so W ′ is a simple neighborhood

of y′ in X ′an and is hence a star-shaped curve. We claim that W ′ = ϕ−1K (W ). Clearly ϕK(W ′) ⊂ W ,

so it suffices to show that the fibers of ϕK have length equal to deg(ϕK). This is certainly the casefor ϕ−1

K (y) = y′. If B ⊂ Xan is the formal fiber of a point in Uk(k), then as in the proof ofProposition 6.15(1), ϕ−1

K (B) ⊂ W ′ is a disjoint union of deg(ϕK) open balls mapping isomorphicallyonto B. For each i the inverse image of S(a)+ ⊂ B(1)+

∼= Bi in W ′ is a disjoint union of annuli, onefor each point in the fiber of ϕ containing x′i, and the degree of ϕK restricted to each annulus is theramification degree of ϕ at the corresponding point. The sum of the ramification degrees of ϕ at thepoints in any fiber is equal to deg(ϕ) = deg(ϕ), which proves the claim. Therefore ϕK induces a finitemorphism W ′ →W , which is a tame covering of (W, y) (really of (W, y, ∅)).

By construction, (Y, y) is a compactification of (W, y), so by Corollary 6.16, the tame coveringW ′ → W lifts to a tame covering Y ′ → Y relative to D. It is clear that (Y ′, y′, ϕ−1

K (D)) is a lifting ofC ′ to a punctured star-shaped curve over (Y, y,D). (One can show that in fact Y ′ ∼= X ′an, althoughthis is not clear a priori.)

6.20.4. It remains to prove (still in the case when Y is proper) that liftings are unique up to uniqueisomorphism. Let ϕK : (Y ′, y′, D′) → (Y, y,D) and ϕ′K : (Y ′′, y′′, D′′) → (Y, y,D) be two liftings ofC ′ to punctured star-shaped curves over (Y, y,D). Then Y ′ and Y ′′ are also proper, hence we mayand do assume that they are the analytic generic fibers of smooth, proper, connected formal curvesX′ and X′′ as in (6.20). Since ϕK(y′) = ϕ′K(y′′) = y, there are unique finite morphisms ϕ : X′ → Xand ϕ′ : X′′ → X extending ϕ and ϕ′, respectively, and ϕk and ϕ′k are identified with ϕ under theisomorphisms Cy′ = X′k

∼= C ′ and Cy′′ = X′′k∼= C ′. Let X ′ and X ′′ be the algebraizations of X′ and X′′,

respectively, and let U ′ = X ′\ϕ−1(D) and U ′′ = X ′′\ϕ′−1(D). Then U ′,U ′′ are finite étale coverings of

U lifting U ′ → Uk, so there is a unique isomorphism ψ : U ′ ∼−→ U ′′ over U . Since Y ′ (resp. Y ′′) is the


analytification of the smooth completion of U ′K (resp. U ′′K), ψK extends uniquely to a Y -isomorphismψK : Y ′

∼−→ Y ′′; we have ψK(y′) = y′′, and ψy′ : Cy′∼−→ Cy′′ is a Cy-isomorphism because U ′k

∼−→ U ′′kis a Uk-isomorphism. Hence ψK is an isomorphism of liftings.

As for uniqueness, suppose that ψK : (Y ′, y′) → (Y ′, y′) is an automorphism of Y ′ as a lifting ofC ′. Then ψK extends uniquely to an X -automorphism ψ : X ′ ∼−→ X ′ that is the identity on X ′k, whichrestricts to a U -automorphism of U ′ that is the identity on U ′k. It follows from the uniqueness of U ′ upto unique isomorphism that ψK is the identity when restricted to U ′K , so since Y ′ is the analytificationof the smooth completion of U ′K , the automorphism ψK is the identity. This concludes the proof forproper Y .

6.20.5. Now suppose that (Y, y) is not proper. Let Y → X be a compactification (6.6) of Y ; we willidentify Y with its image in X. Let D1 be the union of D with a choice of one K-point from everyconnected component of X \ Y . Let ϕ : (X ′, y′, D′1) → (X, y,D1) be a lifting of C ′ to a puncturedstar-shaped curve over (X, y,D1) and let Y ′ = ϕ−1(Y ). Then (Y ′, y′, ϕ−1(D)) is a lifting of C ′ to apunctured star-shaped curve over (Y, y,D).

Let ψ : Y ′ → Y ′ be an automorphism of Y ′ as a lifting of C ′. Since ψ induces the identity mapCy′ → Cy′ on residue curves, ψ takes each connected component of Y ′\y′ to itself. Recall thatX ′\Y ′consists of a disjoint union of closed balls around the points of D′1 \ D′, where D′1 = ϕ−1(D1). Letx′ ∈ D′1 \D′, let B′ be the connected component of X ′ \ y′ containing x′, and let B = ϕ(B′) ⊂ X.By Proposition 6.15(2), we can choose isomorphisms B ∼= B(1)+ and B′ ∼= B(1)+ sending x′ andϕ(x′) to 0, such that the composition

B(1)+∼= B′

ϕ−→ B ∼= B(1)+

is of the form t 7→ tδ. Let A′ = Y ′ ∩ B′ and let A = Y ∩ B = ϕ(A′). The isomorphism B ∼= B(1)+

(resp. B′ ∼= B(1)+) identifies A (resp. A′) with an open annulus S(a)+ (resp. S(a′)+) in B(1)+. Sinceψ|A′ : A′ → A′ is an A-morphism, the composition S(a′)+

∼= A′ → A′ ∼= S(a′)+ is of the formt 7→ ζt, where ζ ∈ R× is a δth root of unity. Therefore ψ|A′ extends uniquely to a B-morphismψ|B′ : B′ → B′ fixing 0. Gluing these morphisms together, we obtain an X-morphism ψ : X ′ → X ′

extending ψ : Y ′ → Y ′. By construction this is an automorphism of X ′ as a lifting of C ′ to X, whichis thus the identity. Therefore ψ : Y ′ → Y ′ is the identity.

Let ϕ′ : (Y ′′, y′′, D′′) → (Y, y,D) be another lifting of C ′ to a punctured star-shaped curve over(Y, y,D). By Corollary 6.16, there exists a compactification Y ′′ → X ′′ of Y ′′ and an extension of ϕ′ toa tame covering ϕ′ : (X ′′, y′′, ϕ′−1(D1))→ (X, y,D1). This is another lifting of C ′ to a punctured star-shaped curve over (X, y,D1), so there is an isomorphism ψ : (X ′, y′)

∼−→ (X ′′, y′′) of liftings. SinceY ′ = ϕ−1(Y ) and Y ′′ = ϕ′

−1(Y ), the isomorphism ψ restricts to an isomorphism (Y ′, y′)

∼−→ (Y ′′, y′′)of liftings. n

Corollary 6.21. With the notation in Theorem 6.18, let (Y ′, y′, D′) → (Y, y,D) be a lifting of C ′ to apunctured star-shaped curve over (Y, y,D). Then the natural homomorphism

AutY (Y ′) −→ AutCy (C ′)

is bijective. If (Y ′′, y′′, D′′) → (Y, y,D) is a second lifting of C ′ to a punctured star-shaped curve over(Y, y,D) then the natural map

IsomY (Y ′′, Y ′) −→ IsomCy (Cy′′ , Cy′)

is bijective.

7. CLASSIFICATION OF LIFTINGS OF HARMONIC MORPHISMS OF METRIZED COMPLEXES

Fix a triangulated punctured curve (X,V ∪D) with skeleton Σ = Σ(X,V ∪D) and let τ : Xan → Σbe the canonical retraction. Throughout this section, we assume that Σ has no loop edges. Letϕ : Σ′ → Σ be a tame covering of metrized complexes of curves. A lifting of Σ′ to a tame coveringof (X,V ∪ D) is a tame covering of triangulated punctured curves ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D)


(see (4.30)) equipped with a Σ-isomorphism ϕ−1(Σ) ∼= Σ′ of metrized complexes of curves. SinceV ′ = ϕ−1(V ) and D′ = ϕ−1(D), we will often denote a lifting simply by X ′. Let ϕ : X ′ → X andϕ′ : X ′′ → X be two liftings of Σ′ and let ψ : X ′

∼−→ X ′′ be an X-isomorphism of curves. Then ψrestricts to a Σ-automorphism

ψ|Σ′ : Σ′ ∼= ϕ−1(Σ)∼−→ ϕ′−1(Σ) ∼= Σ′.

We will consider liftings up to X-isomorphism preserving Σ′, i.e. such that ψΣ′ is the identity, and wewill also consider liftings up to isomorphism as curves over X.

7.1. For every finite vertex x ∈ V , let Σ(x) be the connected component of x in x∪(Σ\V ), let Y (x) =τ−1(Σ(x)), and let D(x) = D ∩ Y (x), as in Example 6.14. Then (Y (x), x,D(x)) is a punctured star-shaped curve. By Proposition 2.22, for every finite vertex x′ ∈ V (Σ′) lying above x the morphism ϕx′ :Cx′ → Cx is finite, tamely ramified, and branched only over the points of Cx corresponding to tangentdirections at x represented by edges of Σ(x). Let ψ(x′) : (Y ′(x′), x′, D(x′)) → (Y (x), x,D(x)) be theunique lifting of Cx′ to a punctured star-shaped curve over (Y (x), x,D(x)) provided by Theorem 6.18and let Σ′(x′) = ψ(x′)−1(Σ(x)) = Σ(Y ′, x′ ∪ D′(x′)). Then Σ′(x′) is canonically identified withthe connected component of x′ in x′ ∪ (Σ′ \ Vf (Σ′)) in such a way that for every edge e′ of Σ′(x′),the point redx′(e

′) ∈ Cx′(k) is identified with the point red(e′) defined in (6.7). This induces anidentification of D′(x′) with ϕ−1(D) ∩ Σ′(x′). Let τx′ be the canonical retraction Y ′(x′) → Σ′(x′)defined in (6.3).

7.2. Let e′ ∈ Ef (Σ′) and e = ϕ(e′). Choose a, a′ ∈ K× with val(a) = `(e) and val(a′) = `(e′). Letx′, y′ be the endpoints of e′. By Proposition 6.15(3), we can choose isomorphisms τ−1(e) ∼= S(a)+

and τ−1x′ (e′) ∼= S(a′)+, τ−1

y′ (e′) ∼= S(a′)+ in such a way that the finite morphisms τ−1x′ (e′)→ τ−1(e)

and τ−1y′ (e′) → τ−1(e) are given by t 7→ tde′ (ϕ), where de′(ϕ) is the degree of the edge map e′ → e.

In particular, there exists a τ−1(e)-isomorphism τ−1x′ (e′)

∼−→ τ−1y′ (e′). As explained in (6.11), there

are canonical identifications

Autτ−1(e)(de′(ϕ)) B Autτ−1(e)(τ−1x′ (e′)) = Autτ−1(e)(τ

−1y′ (e′)) ∼= Z/de′(ϕ)Z;

hence the set of isomorphisms τ−1x′ (e′)

∼−→ τ−1y′ (e′) is a principal homogeneous space under the pre-

or post-composition action of Autτ−1(e)(de′(ϕ)).Let E±f (Σ′) denote the set of oriented finite edges of Σ′, and for e′ ∈ E±f (Σ′) let e′ denote the

same edge with the opposite orientation. Let G(Σ′, X) denote the set of tuples (Θe′)e′∈E±f (Σ′) ofisomorphisms

(7.2.1) Θe′ : τ−1x′ (e′)

∼−→ τ−1y′ (e′)

such that Θe′ = Θ−1e′ , where e′ =

−−→x′y′. We call G(Σ′, X) the set of gluing data for a lifting of Σ′ to a

tame covering of (X,V ∪D), and we emphasize that G(Σ′, X) is nonempty.

7.3. Let α ∈ AutΣ(Σ′), so α is a degree-1 finite harmonic morphism Σ′∼−→ Σ′ preserving Σ′ → Σ.

Let x′ ∈ Vf (Σ′) and let x′′ = α(x′) and x = ϕ(x′) = ϕ(x′′). Part of the data of α is a Cx-isomorphismαx′ : Cx′

∼−→ Cx′′ . By Corollary 6.21 there is a unique lift of αx′ to a Y (x)-isomorphism Y ′(x′)∼−→

Y ′(x′′) of punctured star-shaped curves inducing the isomorphism Cx′∼−→ Cx′′ on residue curves. Let

e′ ∈ Ef (Σ′) be an edge adjacent to x′, let e′′ = α(e′), and let e = ϕ(e′) = ϕ(e′′). Then αx′ restricts toan isomorphism

αx′ : τ−1x′ (e′)

∼−→ τ−1x′′ (e

′′).

Define the conjugation action of AutΣ(Σ′) on G(Σ′, X) by the rule

(7.3.1) α · (Θe′)e′∈E±f (Σ′) = (α−1y′ Θα(e′) αx′)e′∈E±f (Σ′)

where e′ =−−→x′y′.


Theorem 7.4. (Classification of lifts of harmonic morphisms) Let (X,V ∪D) be a triangulated punc-tured curve with skeleton Σ = Σ(X,V ∪D). Assume that Σ has no loop edges. Let ϕ : Σ′ → Σ be a tamecovering of metrized compexes of curves.

(1) There is a canonical bijection between the set of gluing data G(Σ′, X) and the set of liftings of Σ′

to a tame covering of (X,V ∪D), up to X-isomorphism preserving Σ′. In particular, there existsa lifting of Σ′. Any such lifting has no nontrivial automorphisms which preserve Σ′.

(2) Two tuples of gluing data determine X-isomorphic curves if and only if they are in the same orbitunder the conjugation action (7.3.1). The stabilizer in AutΣ(Σ′) of an element of G(Σ′, X) iscanonically isomorphic to the X-automorphism group of the associated curve.

Proof. Given (Θe′) ∈ G(Σ′, X) one can glue the local lifts Y (x′)x′∈Vf (Σ′) via the isomorphisms(7.2.1) to obtain an analytic space which one easily verifies is smooth and proper, hence arises asthe analytification of an algebraic curve X ′. Moreover the morphisms Y ′(x′) → Y (x) glue to give amorphism X ′an → Xan, which is the analytification of a morphism ϕ : X ′ → X. By construction, ifV ′ = ϕ−1(V ) and D′ = ϕ−1(D), then (X ′, V ′ ∪D′)→ (X,V ∪D) is a lifting of Σ′ to a tame coveringof (X,V ∪D).

Now let ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D) be any lifting of Σ′ to a tame covering of (X,V ∪ D).As explained in Example 6.14, for every x ∈ V the inverse image ϕ−1(Y (x)) is a disjoint unionof tame covers of the punctured star-shaped curve (Y (x), x,D(x)), one for each x′ ∈ ϕ−1(x). ByTheorem 6.18, we have canonical identification ϕ−1(Y (x)) =

∐x′ 7→x Y (x′). For x′, y′ ∈ V ′ we have

Y (x′) ∩ Y (y′) =∐e′3x′,y′ τ

−1(e′), where τ : X ′an → Σ′ is the retraction. Hence X ′an is obtained bypasting the local liftings Y (x′) via a choice of isomorphisms (Θe′) ∈ G(Σ′, X).

Let ψ : X ′∼−→ X ′ be an X-automorphism preserving Σ′. Since ψ is the identity on the set

Σ′ ⊂ X ′an, for x′ ∈ V ′ we have ψ(Y (x′)) = Y (x′). We also have that for x′ ∈ V ′ the map of residuecurves ψx′ : Cx′ → Cx′ is the identity, so by Corollary 6.21, ψ restricts to the identity morphismY (x′)→ Y (x′). Since X ′an =

⋃x′∈V ′ Y (x′) we have that ψ is the identity. It follows from this that the

tuple (Θe′) can be recovered from the class of X ′ modulo X-isomorphisms preserving Σ′, so differentgluing data give rise to curves which are not equivalent under such isomorphisms. This proves (1).

Let ϕ : (X ′, V ′ ∪ D′) → (X,V ∪ D) and ϕ′′ : (X ′′, V ′′ ∪ D′′) → (X,V ∪ D) be the liftings of Σ′

associated to the tuples of gluing data (Θ′e′), (Θ′′e′) ∈ G(Σ′, X), respectively. Suppose that there exists

α ∈ AutΣ(Σ′) such that α · (Θ′e′) = (Θ′′e′). Then for all e′ =−−→x′y′ ∈ E±f (Σ′) we have

Θ′′α(e′) αx′ = αy′ Θ′e′ ,

so the Y (ϕ(x′))-isomorphisms αx′ : Y (x′)∼−→ Y (α(x′)) glue to give an X-isomorphism α : X ′

∼−→X ′′. Conversely, the restriction of an X-isomorphism α : X ′

∼−→ X ′′ to Σ′ is a Σ-automorphismof Σ′. It is easy to see that these are inverse constructions. Taking X ′ = X ′′ we have an injectivehomomorphism AutX(X ′) → AutΣ(Σ′); it follows formally from the above considerations that itsimage is the stabilizer of (Θ′e′). n

Remark 7.5. LetX ′ be a lifting of Σ′ to a tame covering of (X,V ∪D). It follows from Theorem 7.4(1)that the natural homomorphism

AutX(X ′) −→ AutΣ(Σ′)

is injective. It is not in general surjective: see Example 7.8 (where all the stabilizer groups of elementsof G(Σ′, X) are proper subgroups of AutΣ(Σ′)).

Remark 7.6. It is worth mentioning that the question of lifting (and classification of all possibleliftings) in the wildly ramified case is more subtle and cannot be guaranteed in general. Some liftingresults in the wildly ramified case are known: e.g., Liu proves in [Liu03, Proposition 5.4] that anyfinite surjective generically étale admissible cover of proper semistable curves over an algebraicallyclosed field k lifts to a finite morphism of smooth proper curves over K. However, the same statementfor metrized complexes cannot be true in general: consider the metrized complex C consisting of asingle finite vertex v and an infinite vertex u attached to v by an infinite edge e with Cv ∼= P1

k and


redv(e) = ∞. Consider the degree p generically étale morphism of metrized complexes C → C whichrestricts to a degree p Artin-Schreier cover ϕ : P1

k → P1k, étale over A1

k → A1k and ramified at∞ (with

ramification index p), and which is linear of slope p on the infinite edge e. This map cannot be liftedto a degree p cover P1

K → P1K in characteristic zero, otherwise, we would obtain a (connected) étale

degree p cover of A1K which is impossible since A1

K is simply connected.

We now state a variant of Theorem 7.4 in which we allow loop edges.

Theorem 7.7. Let (X,V ∪ D) be a triangulated punctured K-curve, let Σ = Σ(X,V ∪ D), and letϕ : Σ′ → Σ be a tame covering of Λ-metrized complexes of k-curves. Then there exists a tame coveringϕ : (X ′, V ′ ∪D′)→ (X,V ∪D) of triangulated punctured K-curves lifting ϕ.

Proof. The only issue is that in Theorem 7.7, we merely require that V is a semistable vertex setwhereas whereas in Theorem 7.4 we require it to be strongly semistable. However, we can modifyΣ by inserting a (valence 2) vertex v along each loop edge, with Cv ∼= P1

k, and with the markedpoints on Cv being 0 and ∞. Call the resulting metrized complex Σ, and let V 0 denote the set ofvertices which have been added to the vertex set V for Σ. We construct a new metrized complexΣ′ from Σ′ by adding V ′0 := ϕ−1(V 0) to the vertex set V ′ for Σ′ and letting C ′v′ ∼= P1

k for v′ ∈ V ′0,with the marked points on C ′v′ being 0 and ∞. We can extend the tame covering ϕ : Σ′ → Σ to atame covering Σ′ → Σ by letting the map from C ′v′ to Cϕ(v′), for v′ ∈ V ′0, be z 7→ zd, where d is thelocal degree of ϕ along the loop edge corresponding to v′. By Theorem 7.4, there is a tame covering(X ′, V ′ ∪ V ′0 ∪ D′) → (X,V ∪ V0 ∪ D) lifting Σ′ → Σ, where V ′0 (resp. V0) corresponds to V

′0 (resp.

V 0). Removing the vertices in V ′0 and V0 gives a tame covering (X ′, V ′ ∪ D′) → (X,V ∪ D) liftingϕ. n

Example 7.8. In this example we suppose that char(k) 6= 2. Let E be a Tate curve over K, let Σ beits (set-theoretic) skeleton, and let τ : Ean → Σ be the canonical retraction. Let U : Gan

m /qZ ∼−→ Ean

be the Tate uniformization of Ean. The 2-torsion subgroup of E is U(±1,±√q); choose a squareroot of q, let y = U(1) and z = U(

√q), and let V = y, z ⊂ Σ. This is a semistable vertex set of

E and Σ = Σ(E, V ) is the circle with circumference val(q); the points y and z are antipodal on Σ.Let e1, e2 be the edges of Σ; orient e1 so that y is the source vertex and e2 so that z is the sourcevertex. The residue curves Cy and Cz are both isomorphic to P1

k; fix isomorphisms Cy ∼= P1k and

Cz ∼= P1k such that the tangent direction at a vertex in the direction of the outgoing (resp. incoming)

edge corresponds to∞ (resp. 0).Let Σ′ be a circle of circumference 1

2 val(q), let V ′ = y′, z′ be a pair of antipodal points on Σ′, and

let e′1 =−−→y′z′ and e′2 =

−−→z′y′ be the two edges of Σ′, with the indicated orientations. Enrich Σ′ with the

structure of a metrized complex of curves by setting Cy′ = Cz′ = P1k and letting the tangent direction

at a vertex in the direction of the outgoing (resp. incoming) edge correspond to∞ (resp. 0). Define amorphism ϕ : Σ′ → Σ of metric graphs by

ϕ(y′) = y, ϕ(z′) = z, ϕ(e′1) = e1, ϕ(e′2) = e2,

with both degrees equal to 2. See Figure 3.Topologically, Σ′ → Σ is a homeomorphism. Let ϕy′ : Cy′ → Cy and ϕz′ : Cz′ → Cz both be the

morphism t 7→ t2 : P1k → P1

k. This makes ϕ into a degree-2 tame covering of metrized complexes ofcurves.

For x ∈ V the analytic space Y (x) is an open annulus of logarithmic modulus val(q); fix an isomor-phism Y (x) ∼= S(q)+ for each x. For x′ ∈ V ′ the analytic space Y ′(x′) is an open annulus of logarithmicmodulus 1

2 val(q); fix isomorphisms Y ′(x′) ∼= S(√q)+ such that the morphisms Y ′(x′)→ Y (ϕ(x′)) are

given by t 7→ t2 : S(√q)+ → S(q)+. For e′ ∈ E(Σ′) we have dϕ(e′) = 2 by definition. Hence G(Σ′, E)

has four elements, which we label (±1,±1). By Theorem 7.4(1) there are four corresponding classesof liftings of Σ′ to a tame covering of (E, V ) up to isomorphism preserving Σ′.

Any Σ-automorphism of Σ′ is the identity on the underlying topological space. Hence an elementof AutΣ(Σ′) is a pair of automorphisms (ψy′ , ψz′) ∈ AutCy (Cy′)×AutCz (Cz′) such that ψy′ , ψz′ fix the


yy′ z′Σ′ zϕ

e′1

e′2

e1

e2

Σ

FIGURE 3. This figure illustrates Example 7.8. The morphism ϕ is a homeomorphismof underlying sets but has a degree of 2 on e′1 and e′2. The arrows on e1, e2, e

′1, e′2

represent the chosen orientations and the indicated tangent vectors at y, z, y′, z′ rep-resent the tangent direction corresponding to∞ in the corresponding k-curve.

points 0,∞ ∈ P1k. Thus ψy′ , ψz′ = ±1, so AutΣ(Σ′) = ±1 × ±1. For x′ = y′, z′ the automorphism

−1 : Cx′∼−→ Cx′ lifts to the automorphism −1 : S(

√q)+

∼−→ S(√q)+. Therefore the conjugation

action of AutΣ(Σ′) on G(Σ′, E) is given as follows:

(1, 1) · (±1,±1) = (−1,−1) · (±1,±1) = (±1,±1)

(−1, 1) · (±1,±1) = (1,−1) · (±1,±1) = −(±1,±1).

By Theorem 7.4(2), there are two isomorphism classes of lifts of Σ′ to a tame covering of (E, V ), andeach such lift has two automorphisms.

These liftings can be described concretely as follows. Fix a square root of q, let E± be the alge-braization of the analytic elliptic curve Gan

m /(±√q)Z, and let ψ± : E± → E be the morphism t 7→ t2

on uniformizations. Let Σ± = ψ−1± (Σ). Then Σ± is isomorphic to Σ′ as a tame covering of Σ and E± is

a lifting of Σ′ to a tame covering of (E,D). The elliptic curves E± are not isomorphic (as K-schemes)because they have different q-invariants, so they represent the two isomorphism classes of liftings ofΣ′. The nontrivial automorphism of E± is given by translating by the image of −1 ∈ Gan

m (K) (this isnot a homomorphism). In fact, since ψ± : E± → E is an étale Galois cover of degree 2, this is the onlynontrivial automorphism of E± as an E-curve, so the homomorphism

AutE(E±) −→ AutΣ(Σ′) ∼= ±1 × ±1

is injective but not surjective (its image is ±(1, 1)).

7.9. Consider now the automorphism group Aut0Σ(Σ′) ⊂ AutΣ(Σ′) consisting of all degree-1 finite

harmonic morphisms α : Σ′ → Σ′ respecting ϕ : Σ′ → Σ and inducing the identity on the metric graphΓ′ underlying Σ′. The restriction of the conjugacy action of AutΣ(Σ′) on G(Σ′, X) to the subgroupAut0

Σ(Σ′) admits a simplified description that we describe now. Combining this with arguments similarto those in the proof of Theorem 7.4 yields a classification of the set of liftings of Σ′ up to isomorphismas liftings of the metric graph underlying Σ′; see Theorem 7.10.

First, for each x′ ∈ V (Σ′) with image x = ϕ(x′), let Aut0Cx(Cx′) be the subgroup of AutCx(Cx′)

that fixes every point of Cx′ of the form red(e′) for some edge e′ of Σ′ adjacent to x′. Then

Aut0Σ(Σ′) =

∏x′∈Vf (Σ′)

Aut0Cϕ(x)

(Cx′) =: E0.

Denote by E1 the finite abelian group

E1 =∏

e′∈Ef (Σ′)

Autτ−1(ϕ(e′))(de′(ϕ)).

The discussion preceding Theorem 7.4 shows that the set of gluing data G(Σ′, X) is canonically aprincipal homogeneous space under E1.


The subgroup Aut0Cx(Cx′) ⊂ AutCx(Cx′) corresponds to the subgroup Aut0

Y (x)(Y′(x′)) of au-

tomorphisms in AutY (x)(Y′(x′)) which act trivially on the skeleton Σ′(x′). Restriction of a Y (x)-

automorphism of Y ′(x′) to τ−1x′ (e′) defines a homomorphism

ρx′,e′ : Aut0Cx(Cx′) = Aut0

Y (x)(Y′(x′)) −→ Autτ−1(e)(de′(ϕ)).

Fix an orientation of each finite edge of Σ′. For x′ ∈ Vf (Σ′) with image x = ϕ(x′), let ρx′ :

Aut0Cx(Cx′)→ E1 be the homomorphism whose e′-coordinate is

(ρx′(α))e′ =

ρx′,e′(α) if x′ is the source vertex of e′

ρx′,e′(α)−1 if x′ is the target vertex of e′

1 if x′ is not an endpoint of e′.

Taking the product over all x′ ∈ Vf (Σ′) yields a homomorphism ρ : E0 → E1. The kernel and cokernelof ρ are independent of the choice of orientations. Viewing E0 → E1 as a two-term complex E• ofgroups, its cohomology groups are

H0(E•) = ker(ρ) and H1(E•) = coker(ρ).

Theorem 7.10. Let (X,V ∪ D) be a triangulated punctured curve with skeleton Σ = Σ(X,V ∪ D).Assume that Σ has no loop edges. Let ϕ : Σ′ → Σ be a tame covering of metrized compexes of curves.

(1) G(Σ′, X) is canonically a principal homogeneous space under E1 and the conjugacy action ofAut0

Σ(Σ′) on G(Σ′, X) is given by the action of the subgroup ρ(Aut0Σ(Σ′)) = ρ(E0) ⊆ E1 on

G(Σ′, X).(2) The set of liftings of Σ′ up to isomorphism as liftings of the metric graph underlying Σ′ is a

principal homogeneous space under H1(E•), and the group of automorphisms of a given liftingas a lifting of the metric graph underlying Σ′ is isomorphic to H0(E•).

7.11. Descent to a general ground field. Let K0 be a subfield of K, let X0 be a smooth, projective,geometrically connected K0-curve, and let D ⊂ X0(K0) be a finite set. Let X = X0 ⊗K0

K, let V be astrongly semistable vertex set of (X,D), let Σ = Σ(X,V ∪D), and let ϕ : Σ′ → Σ be a tame coveringof metrized complexes of curves, as in the statement of Theorem 7.4. Let ϕ : X ′ → X be a lifting of ϕto a tame covering of (X,V ∪D). Whereas we take the data of the morphism Σ′ → Σ to be geometric,i.e. only defined over K, the covering X ′ → X is in fact defined over a finite, separable extension ofK0. This follows from the fact that if U = X \D and U ′ = X ′ \ ϕ−1(D), then ϕ : U ′ → U is a tamelyramified cover of U over X relative to D (see Remarks 2.20(1) and 4.32(1)), along with the followinglemma.

Lemma 7.12. Let K0 be any field, let K be a separably closed field containing K0, let X0 be a smooth,projective, geometrically connected K0-curve, and let D ⊂ X0(K0) be a finite set. Let X = X0⊗K0 K, letϕ : X ′ → X be a finite morphism with X ′ smooth and (geometrically) connected, and suppose that ϕ isbranched only over D, with all ramification degrees prime to the characteristic of K. Then there exists afinite, separable extension K1 of K and a morphism ϕ1 : X ′1 → X0 ⊗K0

K1 descending ϕ.Proof. Let U0 = X0\D, and let U = X\D and U ′ = X ′\ϕ−1(D). First suppose thatK0 is separably

closed. By [SGA1, Exposé XIII, Corollaire 2.12], the tamely ramified étale fundamental groups πt1(U0)and πt1(U) are isomorphic (with respect to some choice of base point). Since ϕ : U ′ → U is a tamelyramified cover of U over X relative to D, it is classified by a finite-index subgroup of πt1(U) = πt1(U0),so there exists a tamely ramified cover ϕ0 : U ′0 → U0 of U0 over X0 relative to D descending ϕ.

Now we drop the hypothesis thatK0 is separably closed. By the previous paragraph we may assumethat K is a separable closure of K0. By general principles the projective morphism X ′ → X descendsto a subfield of K which is finitely generated (i.e. finite) over K0. n

Remark 7.13. With the notation in (7.11), suppose that K0 is a complete valued field with valuegroup Λ0 = val(K×0 ) and algebraically closed residue field k, that Σ is “rational over K0” in thatit comes from a (split) semistable formal R0-model of X0 in the sense of Section 5, and that Σ′ has


edge lengths contained in Λ0. With some extra work it is possible to carry out the gluing argumentsof Theorem 7.7 directly over the field K0 (in this context Lemma 6.10(1) still holds), which showsthat the cover X ′ → X is in fact defined over K0. In the case of a discrete valuation this also followsfrom [Wew99], or from [Saï97] if D = ∅.7.14. Liftings of tame harmonic morphisms. Theorem 7.7 implies the existing of liftings for (fi-nite) tame harmonic morphisms of metrized complexes which are not necessarily tame coverings, thedifference being generic étaleness. See Definition 2.21 for both definitions.

Proposition 7.15. Let (X,V ∪ D) be a triangulated punctured K-curve, let Σ = Σ(X,V ∪ D), andlet ϕ : Σ′ → Σ be a tame harmonic morphism of metrized complexes of curves. Then there exists atriangulated punctured K-curve (X ′, V ′ ∪ D′) and a finite morphism ψ : (X ′, V ′ ∪ D′) → (X,V ∪ D)lifting ϕ.

Proof. For any finite vertex x′ of Σ′ at which ϕ, seen as a morphism of augmented metric graphs,is ramified, let q1, . . . , qr ∈ Cv be all the branch points of ϕx′ which do not correspond to any edge ofΣ, and let pij denote the preimages of qi under ϕx′ . We modify Σ′ and Σ by attaching infinite edges eito Σ at x = ϕ(x′) for each qi, infinite edges e′ij to Σ′ at x′ for each pij , and defining redx(ei) = qi andredx′(e

′ij) = pij . Since each ϕx′ is tamely ramified, the harmonic morphism ϕ naturally extends to a

tame covering ϕ : Σ′ → Σ between the resulting modifications. Enlarging D to D by choosing pointsin X(K) \D with reduction qi, we can assume that Σ = Σ(X,V ∪ D). The result now follows fromTheorem 7.7 by first lifting ϕ to a tame covering ψ : (X ′, V ′ ∪ D′)→ (X,V ∪ D) and then taking therestriction of ψ to (X ′, V ′ ∪D′)→ (X,V ∪D), where D′ = ψ−1(D). n

Remark 7.16. As mentioned above, in Proposition 7.15 we do not require that ϕ : Σ′ → Σ begenerically étale. This corresponds to not requiring that D ⊂ X(K) contain the branch locus of thelift ψ : X ′ → X: indeed, by Lemma 4.33, if D contains the branch locus then ϕ is a tame covering. Inthe situation of Proposition 7.15 the set of liftings of Σ′ to a tame covering of (X,V ∪D) can be infinite.For example, if Γ′ = p′ and Γ = p are both points and the morphism ϕp′ : Cp′ ∼= P1 → Cp ∼= P1

is z 7→ z2, with (X,V ) a minimal triangulation of P1 and D = ∅, then there are infinitely many suchlifts, corresponding to the different ways of lifting the critical points and critical values of ϕp′ from kto K.

8. APPLICATION: COMPONENT GROUPS OF NÉRON MODELS

8.1. In contrast to the rest of the paper, we assume in this section that R is a complete discretevaluation ring with fraction field K and algebraically closed residue field k. In this case we take thevalue group Λ to be Z.

There is a natural notion of harmonic 1-forms on a metric graph Γ of genus g (see [MZ08]). Thespace Ω1(Γ) is a g-dimensional real vector space which can be canonically identified with H1(Γ,R),but we write elements of Ω1(Γ) as ω =

∑e ωe de as in [BF11], where the sum is over all edges with

respect to a fixed vertex set for Γ. There is a canonical lattice Ω1Z(Γ) of integer harmonic 1-forms

inside Ω1(Γ); these are the harmonic 1-forms for which every ωe is an integer. A harmonic morphismϕ : Γ′ → Γ induces a natural pullback map on harmonic 1-forms via the formula

ϕ∗(∑e

ωe de) =∑e′

de′(ϕ) · ωϕ(e′) de′.

Recall that a Z-metric graph (i.e. a Λ-metric graph for Λ = Z) with no infinite vertices is a (compactand finite) metric graph whose edge lengths are all positive integers (or equivalently, having a vertexset with respect to which all edge lengths are 1). If X/K is a smooth, proper, geometrically connectedanalytic curve and X is a semistable R-model forX, the skeleton ΓX of X is naturally a Z-metric graph.Moreover, as we have seen earlier in this paper, a finite morphism of semistable models induces in anatural way a harmonic morphism of Z-metric graphs.


Let Γ be a Z-metric graph with no infinite vertices. We define the regularized Jacobian Jacreg(Γ) ofΓ to be the group Jac(G), where G is the “regular model” for Γ (the unique graph induced by a vertexset with all edge lengths equal to 1). The group Jac(G) can be described explicitly as

Jac(G) = Ω1(G)#/H1(G,Z),

where Ω1(G)# denotes the linear functionals Ω1(G,R) → R of the form∫α

with α ∈ C1(G,Z) (see[BF11]). There is a canonical isomorphism between Jac(G) and Pic0(G), the group of divisors ofdegree 0 on G modulo the principal divisors (see [BdlHN97]) as well as a canonical isomorphism

Jac(G) ∼= H1(G,Z)#/Ω1Z(G),

where

H1(G,Z)# =

ω ∈ Ω1(G,R) :

∫γ

ω ∈ Z ∀γ ∈ H1(G,Z)

.

We recall (in our own terminology) the following result of Raynaud (cf. [Ray70] and [Bak08,Appendix A]):

Theorem 8.2. (Raynaud) If X/K is a semistable curve, then the component group of the Néron modelof Jac(X) over R is canonically isomorphic to Jacreg(ΓX) (for any semistable model X of X).

A harmonic morphism ϕ : Γ′ → Γ of Z-metric graphs induces in a functorial way homomorphismsϕ∗ : Jacreg(Γ′)→ Jacreg(Γ), defined by

ϕ∗

([∫α′

])=

[ ∫ϕ∗(α′)

],

whereϕ∗(∑

αe′e′) =

∑αe′ϕ(e′),

and ϕ∗ : Jacreg(Γ)→ Jacreg(Γ′), defined by

ϕ∗([ω]) = [ϕ∗ω],

whereϕ∗(∑

ωede)

=∑e′

de′(ϕ)ωϕ(e′)de′.

We have the following elementary result, whose proof we omit:

Lemma 8.3. Under the canonical isomorphism between Jac(G) and Pic0(G), the homomorphismϕ∗ : Jac(G′) → Jac(G) corresponds to the map [D′] 7→ [ϕ∗(D

′)] from Pic0(G′) to Pic0(G), whereϕ∗(∑av′(v

′)) =∑av′ϕ(v′).

The following is a “relative” version of Raynaud’s theorem; it implies that the covariant functorwhich takes a semistable R-model X for a K-curve X to the component group of the Néron modelof the Jacobian of X factors as the “reduction graph functor” X 7→ ΓX followed by the “regularizedJacobian functor” Γ 7→ Jacreg(Γ). It is a straightforward consequence of the analytic description ofRaynaud’s theorem given in [Bak08, Appendix A] (see also [BR13]):

Theorem 8.4. If fK : X ′ → X is a finite morphism of curves over K, the induced maps f∗ :ΦJ(X′) → ΦJ(X) and f∗ : ΦJ(X) → ΦJ(X′) on component groups coincide with the induced mapsϕ∗ : Jacreg(ΓX′) → Jacreg(ΓX) and ϕ∗ : Jacreg(ΓX) → Jacreg(ΓX′) for any morphism f : X′ → X ofsemistable models extending fK . (Here ϕ denotes the harmonic morphism of skeleta induced by f ; seeRemark 5.14.)

It follows easily from Lemma 8.3 and Theorem 8.4 that if fK : X ′ → X is regularizable, i.e., iffK extends to a morphism of regular semistable models, then the induced map f∗ : ΦX′ → ΦX oncomponent groups is surjective. Thus whenever f∗ is not surjective, it follows that fK does not extendto a morphism of regular semistable models. One can obtain a number of concrete examples of this


situation from modular curves (e.g. the map X0(33) → E over Qunr3 , where E is the optimal elliptic

curve of level 33.)

Remark 8.5. One can show that if ϕ : Γ′ → Γ is a harmonic morphism of Z-metric graphs, thenϕ∗ : ΦX′ → ΦX is surjective iff ϕ∗ : ΦX → ΦX′ is injective. Indeed, it is not hard to check that themaps ϕ∗ and ϕ∗ are adjoint with respect to the combinatorial monodromy pairing, the non-degeneratesymmetric bilinear form 〈 , 〉 : Jac(G) × Jac(G) → Q/Z defined by 〈[ω], [

∫α

]〉 = [∫αω], where

ω ∈ H1(G,Z)# and∫α∈ Ω1(G)#. Thus the groups ker(ϕ∗) and coker(ϕ∗) are canonically dual. This

is a combinatorial analogue of results proved by Grothendieck in SGA7 on the (usual) monodromypairing.

As an application of Theorem 8.4 and our results on lifting harmonic morphisms, one can constructmany examples of harmonic morphisms of Z-metric graphs for which ϕ∗ is not surjective. For example,consider the following question posed by Ken Ribet in a 2007 email correspondence with the secondauthor (Baker):

Suppose f : X ′ → X is a finite morphism of semistable curves over a complete discretelyvalued field K with g(X) ≥ 2. Assume that the special fiber of the minimal regular model ofX ′ consists of two projective lines intersecting transversely. Is the induced map f∗ : ΦX′ →ΦX on component groups of Néron models necessarily surjective?

We now show that the answer to Ribet’s question is no.

Example 8.6. Consider the “banana graph” B(`1, . . . , `g+1) consisting of two vertices and g + 1edges of length `i. This is the reduction graph of a semistable curve whose reduction has two P1’scrossing transversely at singular points of thickness `1, . . . , `g+1. If we set G′ = B(1, 1, 1, 1) andG = B(1, 2, 2) and let Γ′,Γ be the geometric realizations of G′ and G, respectively, then there is adegree 2 harmonic morphism of Z-metric graphs ϕ : Γ′ → Γ taking e′1 and e′2 to e1, e′3 to e2, and e′4to e3. The homomorphism ϕ∗ is non-surjective since | Jacreg(Γ)| = 4 and | Jacreg(Γ′)| = 8. The map ϕcan be enriched to a harmonic morphism ϕ of metrized complexes of curves by attaching a P1 to eachvertex and letting each morphism P1 → P1 be z 7→ z2, with 0, ±1, and ∞ being the marked pointsupstairs. By Remark 7.13, the morphism ϕ lifts to a morphism ψ : X ′ → X of curves over K. Sinceall edges of G′ have length 1, the minimal proper regular model of X ′ has a special fiber consisting oftwo P1’s crossing transversely. By Theorem 8.4, the induced covariant map on component groups ofNéron models is not surjective.

REFERENCES

[AB12] O. Amini and M. Baker, Linear series on metrized complexes of algebraic curves, Preprint available at http://arxiv.org/abs/1204.3508.

[ABBR14] O. Amini, M. Baker, E. Brugallé, and J Rabinoff, Lifting harmonic morphisms II: tropical curves and metrized com-plexes, http://arxiv.org/abs/1404.3390 (2014).

[ACP12] S. Abramovich, L. Caporaso, and S. Payne, The tropicalization of the moduli space of curves, Preprint available athttp://arxiv.org/abs/1212.0373.

[Ana00] C.K. Anand, A discrete analogue of the harmonic morphism, Harmonic morphisms, harmonic maps, and relatedtopics (Brest, 1997), Chapman & Hall/CRC Res. Notes Math., vol. 413, Chapman & Hall/CRC, Boca Raton, FL,2000, pp. 109–112.

[Bak08] M. Baker, Specialization of linear systems from curves to graphs, Algebra Number Theory 2 (2008), no. 6, 613–653,With an appendix by Brian Conrad.

[BBM11] B. Bertrand, E. Brugallé, and G. Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125(2011), 157–171.

[BdlHN97] R. Bacher, P. de la Harpe, and T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finitegraph, Bull. Soc. Math. France 125 (1997), no. 2, 167–198.

[Ber90] V. G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys andMonographs, vol. 33, American Mathematical Society, Providence, RI, 1990.

[Ber93] , Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. 78 (1993),5–161.

[Ber99] , Smooth p-adic analytic spaces are locally contractible, Invent. Math. 137 (1999), no. 1, 1–84.


[BF11] M. Baker and X. Faber, Metric properties of the tropical Abel-Jacobi map, J. Algebraic Combin. 33 (2011), no. 3,349–381.

[BGR84] S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984.

[BL85] S. Bosch and W. Lütkebohmert, Stable reduction and uniformization of abelian varieties. I, Math. Ann. 270 (1985),no. 3, 349–379.

[BL93] , Formal and rigid geometry. I. Rigid spaces, Math. Ann. 295 (1993), no. 2, 291–317.[BN07] M. Baker and S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph, Adv. Math. 215 (2007), no. 1,

766–788.[BN09] , Harmonic morphisms and hyperelliptic graphs, International Math. Research Notices 15 (2009), 2914–

2955.[BPR11] M. Baker, S. Payne, and J. Rabinoff, Non-Archimedean geometry, tropicalization, and metrics on curves, 2011, Preprint

available at http://arxiv.org/abs/1104.0320.[BR10] M. Baker and R. Rumely, Potential theory and dynamics on the Berkovich projective line, Mathematical Surveys and

Monographs, vol. 159, American Mathematical Society, Providence, RI, 2010.[BR13] M. Baker and J. Rabinoff, An analytic version of Raynaud’s description of component groups of Néron models, in

preparation (2013).[Cha12] M. Chan, Tropical hyperelliptic curves, J. Alg. Combin. 15 (2012), 2914–2955.[CKK] G. Cornelissen, F. Kato, and J. Kool, A combinatorial Li-Yau inequality and rational points on curves, To appear in

Math. Ann., Preprint available at http://arxiv.org/abs/1211.2681.[Col03] R. F. Coleman, Stable maps of curves, Documenta Math. Extra volume Kato (2003), 217–225.[Con99] B. Conrad, Irreducible components of rigid spaces, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 473–541.[Duc08] A. Ducros, Triangulations et cohomologie étale sur une courbe analytique p-adique, J. Algebraic Geom. 17 (2008),

no. 3, 503–575.[Fab13a] X. Faber, Topology and geometry of the Berkovich ramification locus for rational functions, Manuscripta Math. (2013),

439–474.[Fab13b] , Topology and geometry of the Berkovich ramification locus for rational functions, II, Matt. Ann. 356 (2013),

no. 3, 819–844.[HM82] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1,

23–88, With an appendix by W. Fulton.[KS01] M. Kontsevich and Y. Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror

symmetry (Seoul, 2000), World Sci. Publishing, River Edge, NJ, 2001, pp. 203–263.[KS06] , Affine structures and non-Archimedean analytic spaces, The unity of mathematics, Progr. Math., vol. 244,

Birkhäuser Boston, Boston, MA, 2006, pp. 321–385.[Liu03] Q. Liu, Reduction and lifting of finite covers of curves, Proceedings of the 2003 Workshop on Cryptography and

Related Mathematics, Chuo University, 2003, pp. 161–180.[Liu06] , Stable reduction of finite covers of curves, Compositio Math. 142 (2006), no. 1, 101–118.[LL99] Q. Liu and D. Lorenzini, Models of curves and finite covers, Compositio Math. 118 (1999), no. 1, 61–102.[Mik] G. Mikhalkin, Phase-tropical curves I. Realizability and enumeration, In preparation.[Mik05] , Enumerative tropical algebraic geometry in R2, J. Amer. Math. Soc. 18 (2005), no. 2, 313–377.[Mik06] , Tropical geometry and its applications, International Congress of Mathematicians. Vol. II, Eur. Math. Soc.,

Zürich, 2006, pp. 827–852.[MZ08] G. Mikhalkin and I. Zharkov, Tropical curves, their Jacobians and theta functions, Curves and abelian varieties,

Contemp. Math., vol. 465, Amer. Math. Soc., Providence, RI, 2008, pp. 203–230.[Ray70] M. Raynaud, Spécialisation du foncteur de Picard, Inst. Hautes Études Sci. Publ. Math. (1970), no. 38, 27–76.[Saï96] M. Saïdi, Revêtements étales abéliens, courants sur les graphes et réductions semi-stable des courbes, Manuscripta

Mathematica 89 (1996), 245–265.[Saï97] , Revêtements modérés et groupe fondamental de graphe de groupes, Compos. Math. 107 (1997), 319–338.[Saï04] , Wild ramification and a vanishing cycles formula, J. Algebra 273 (2004), no. 1, 108–128.[SGA1] A. Grothendieck and M. Raynaud, Revêtements étales et groupe fondamental (SGA 1), Séminaire de Géométrie

Algébrique de l’Institut des Hautes Études Scientifiques (SGA 1), Société Mathématique de France, Paris, 2003.[Tem] M. Temkin, Introduction to Berkovich analytic spaces, Preprint available at http://arxiv.org/abs/1010.

2235v1.[Thu05] A. Thuillier, Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. Applications à la

théorie d’Arakelov, Ph.D. thesis, University of Rennes, 2005, Preprint available at http://tel.ccsd.cnrs.fr/documents/archives0/00/01/09/90/index.html.

[Ura00] H. Urakawa, A discrete analogue of the harmonic morphism and green kernel comparison theorems, Glasg. Math. J.42 (2000), 319–334.

[Wel13] J. Welliaveetil, A Riemann-Hurwitz formula for skeleta in non-Archimedean geometry, 2013, Preprint available athttp://arxiv.org/abs/1303.0164.

[Wew99] S. Wewers, Deformation of tame admissible covers of curves, Aspects of Galois theory (Gainesville, FL, 1996), LondonMath. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 239–282.


E-mail address: [email protected]

CNRS-DMA, ÉCOLE NORMALE SUPÉRIEURE, 45 RUE D’ULM, PARIS


SCHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY, ATLANTA GA 30332-0160, USA


UNIVERSITÉ PIERRE ET MARIE CURIE, PARIS 6, 4 PLACE JUSSIEU, 75 005 PARIS, FRANCE


SCHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY, ATLANTA GA 30332-0160, USA

Documents

LIFTING HARMONIC MORPHISMS I: METRIZED COMPLEXES … · LIFTING HARMONIC MORPHISMS I: METRIZED COMPLEXES AND BERKOVICH SKELETA OMID AMINI, MATTHEW BAKER, ERWAN BRUGALLÉ, AND JOSEPH